This document discusses several methods for solving separable differential equations:
- Separable differential equations can be solved by separating the variables and integrating both sides to obtain an implicit solution. The initial condition is then applied to determine any constants.
- Explicit solutions are obtained by solving the implicit solution for the variable y(x).
- The interval of validity, where the solution is defined, is determined by avoiding values of x that would cause issues like division by zero.
- Three examples demonstrate these steps to solve initial value problems involving separable differential equations and determine the interval of validity.
This document discusses intervals of validity for solutions to differential equations. It presents two theorems, one for linear first-order equations and one for non-linear equations. The linear theorem guarantees a unique solution over intervals where coefficients are continuous, allowing validity intervals to be found without solving the equation. However, for non-linear equations the solution must be found to determine the validity interval, which can depend on the initial condition. Examples illustrate finding validity intervals, the possibility of multiple solutions if theorem conditions are violated, and how validity intervals can depend on initial conditions for non-linear equations.
The document discusses solving systems of two and three equations with two or three variables. For two equations, the solutions can be one point (consistent), a line (dependent), or no solution (inconsistent). Graphically, these correspond to lines intersecting at a point, coinciding, or being parallel. The document demonstrates solving two equations algebraically using substitution and elimination. For three equations, the solutions are a point, line, or no solution, corresponding to planes intersecting at a point, along a line, or not intersecting. Elimination is used to solve three equations by eliminating variables to obtain unique or dependent solutions. Examples are worked through to illustrate the different cases.
Modelling with first order differential equationsTarun Gehlot
The document discusses modelling physical situations using first order differential equations. It provides examples of modelling mixing problems, population problems, and falling bodies. Mixing problems involve substances dissolving in liquids entering and leaving tanks, and assume uniform concentration. Population problems use differential equations to model population growth or decline based on birth, death, migration rates. The examples solve related initial value problems and differential equations to determine outcomes like amount of substance in a tank or population size over time.
- The document discusses different types of first order ordinary differential equations (ODEs) including separable, homogeneous, exact, and linear equations.
- It provides examples of identifying each type of equation and the general methods for solving them, such as using separation of variables, substitution to make equations homogeneous or separable, finding integrating factors, and determining if equations are exact.
- Various examples are worked through step-by-step to illustrate each problem solving technique. Exercises are also provided for students to practice applying the methods.
This document provides an overview and examples of Taylor series. Taylor series approximate functions as infinite sums involving derivatives of the function evaluated at a specific point. The document reviews the formula for Taylor series and provides examples of determining Taylor series for common functions like cosine, sine and exponential functions. It demonstrates how to find Taylor series about different points and discusses cases where the Taylor series may terminate after a finite number of terms.
The document discusses the concepts of ill-conditioned and well-conditioned systems of equations. An ill-conditioned system is one where small changes to the coefficients or right-hand side result in large changes to the solution. A well-conditioned system exhibits small changes to the solution given small changes to the inputs. The norm of a matrix and its relationship to the condition number are introduced as a way to quantify how many significant digits can be trusted in a solution. Examples demonstrate calculating the condition number and using it to determine the number of significant digits correct in a solution.
This document provides an overview of topics covered in a differential equations course, including:
1. Review of integration by parts and partial fractions.
2. Discussion of integral curves and the existence and uniqueness theorem for differential equations.
3. Classification and methods for solving first and higher order linear differential equations, including separable, exact, integrating factors, Bernoulli, homogeneous with constant coefficients, and undetermined coefficients.
4. Brief introduction to additional solution methods like Euler's method, power series, and Laplace transforms.
5. Mention of solving systems of linear differential equations.
This document discusses intervals of validity for solutions to differential equations. It presents two theorems, one for linear first-order equations and one for non-linear equations. The linear theorem guarantees a unique solution over intervals where coefficients are continuous, allowing validity intervals to be found without solving the equation. However, for non-linear equations the solution must be found to determine the validity interval, which can depend on the initial condition. Examples illustrate finding validity intervals, the possibility of multiple solutions if theorem conditions are violated, and how validity intervals can depend on initial conditions for non-linear equations.
The document discusses solving systems of two and three equations with two or three variables. For two equations, the solutions can be one point (consistent), a line (dependent), or no solution (inconsistent). Graphically, these correspond to lines intersecting at a point, coinciding, or being parallel. The document demonstrates solving two equations algebraically using substitution and elimination. For three equations, the solutions are a point, line, or no solution, corresponding to planes intersecting at a point, along a line, or not intersecting. Elimination is used to solve three equations by eliminating variables to obtain unique or dependent solutions. Examples are worked through to illustrate the different cases.
Modelling with first order differential equationsTarun Gehlot
The document discusses modelling physical situations using first order differential equations. It provides examples of modelling mixing problems, population problems, and falling bodies. Mixing problems involve substances dissolving in liquids entering and leaving tanks, and assume uniform concentration. Population problems use differential equations to model population growth or decline based on birth, death, migration rates. The examples solve related initial value problems and differential equations to determine outcomes like amount of substance in a tank or population size over time.
- The document discusses different types of first order ordinary differential equations (ODEs) including separable, homogeneous, exact, and linear equations.
- It provides examples of identifying each type of equation and the general methods for solving them, such as using separation of variables, substitution to make equations homogeneous or separable, finding integrating factors, and determining if equations are exact.
- Various examples are worked through step-by-step to illustrate each problem solving technique. Exercises are also provided for students to practice applying the methods.
This document provides an overview and examples of Taylor series. Taylor series approximate functions as infinite sums involving derivatives of the function evaluated at a specific point. The document reviews the formula for Taylor series and provides examples of determining Taylor series for common functions like cosine, sine and exponential functions. It demonstrates how to find Taylor series about different points and discusses cases where the Taylor series may terminate after a finite number of terms.
The document discusses the concepts of ill-conditioned and well-conditioned systems of equations. An ill-conditioned system is one where small changes to the coefficients or right-hand side result in large changes to the solution. A well-conditioned system exhibits small changes to the solution given small changes to the inputs. The norm of a matrix and its relationship to the condition number are introduced as a way to quantify how many significant digits can be trusted in a solution. Examples demonstrate calculating the condition number and using it to determine the number of significant digits correct in a solution.
This document provides an overview of topics covered in a differential equations course, including:
1. Review of integration by parts and partial fractions.
2. Discussion of integral curves and the existence and uniqueness theorem for differential equations.
3. Classification and methods for solving first and higher order linear differential equations, including separable, exact, integrating factors, Bernoulli, homogeneous with constant coefficients, and undetermined coefficients.
4. Brief introduction to additional solution methods like Euler's method, power series, and Laplace transforms.
5. Mention of solving systems of linear differential equations.
1) The document discusses fractions and rational numbers. It notes that US students struggle with fraction arithmetic compared to international peers and provides an example problem where over 42% of US students chose the wrong answer.
2) It advocates understanding fractions as sums of unit fractions (e.g. 2/7 = 1/4 + 1/28) in the same way place value is taught for decimals. This conceptual understanding enables computational fluency with fractions.
3) It distinguishes between fractions, which are written symbols, and rational numbers, which depend on value not notation. A rational number can always be expressed as a quotient of two integers.
This document provides an overview of key concepts in polynomial functions. It defines polynomials as functions where the exponents are positive whole numbers. It explains standard form, factored form, degrees, leading coefficients, roots, ends, and graphing behaviors. It also covers dividing polynomials using long division and synthetic division, and defines the remainder and factor theorems. Specifically, the remainder theorem states the remainder of dividing a polynomial f(x) by (x-k) is r=f(k), while the factor theorem says a polynomial f(x) has a factor (x-k) if and only if f(k)=0.
Exact & non exact differential equationAlaminMdBabu
This document discusses exact and non-exact differential equations. It states that an exact differential equation can be obtained directly by differentiation of its solution, without any other operations. It provides the working rule to check if a differential equation is exact by seeing if the partial derivatives are equal. For non-exact equations, an integrating factor can be found using four different rules depending on the form of the differential equation.
This document provides an overview of Chapter 1 from the textbook "Differential Equations & Linear Algebra" which covers first-order ordinary differential equations. It defines differential equations and their order, provides examples of common types of differential equations and mathematical models, and explains concepts like general/particular solutions and initial value problems. The chapter then covers methods for solving first-order differential equations, including those that are separable, linear, or may require a substitution to transform into a separable or linear equation like the homogeneous or Bernoulli equations. Suggested practice problems are marked for exam inspiration.
1) Linear and nonlinear first order differential equations differ in their theory of existence and uniqueness of solutions, properties of general solutions, and whether solutions can be expressed explicitly or only implicitly.
2) For linear equations, solutions are guaranteed to exist and be unique based on continuity of coefficients, and general solutions involving arbitrary constants can represent all solutions.
3) Nonlinear equations may only have local existence and uniqueness depending on initial conditions, general solutions may not encompass all solutions, and solutions are often only implicitly defined requiring numerical methods.
Oscillation and Convengence Properties of Second Order Nonlinear Neutral Dela...inventionjournals
In this paper, we consider the second order nonlinear neutral delay difference equations of the form We establish sufficient conditions which ensures that every solution of is either oscillatory or tends to zero as . We also gives examples to illustrate our results
Formal expansion method for solving an electrical circuit modelTELKOMNIKA JOURNAL
We investigate the validity of the formal expansion method for solving a second order ordinary differential equation raised from an electrical circuit problem. The formal expansion method approximates the exact solution using a series of solutions. An approximate formal expansion solution is a truncated version of this series. In this paper, we confirm using simulations that the approximate formal expansion solution is valid for a specific interval of domain of the free variable. The accuracy of the formal expansion approximation is guaranteed on the time-scale 1.
This document discusses solving systems of linear equations with two or three variables. It explains that a system can have one solution, infinitely many solutions, or no solution. For two variables, the solutions are where lines intersect (one solution), coincide (infinitely many), or are parallel (no solution). For three variables, the solutions are where planes intersect (one point), lie on a line (infinitely many), or do not intersect (no solution). It demonstrates solving systems using substitution, elimination, and matrix methods, and discusses cases where a system has infinitely many or no solutions.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This chapter discusses classification methods including linear discriminant functions and probabilistic generative and discriminative models. It covers linear decision boundaries, perceptrons, Fisher's linear discriminant, logistic regression, and the use of sigmoid and softmax activation functions. The key points are:
1) Classification involves dividing the input space into decision regions using linear or nonlinear boundaries.
2) Perceptrons and Fisher's linear discriminant find linear decision boundaries by updating weights to minimize misclassification.
3) Generative models like naive Bayes estimate joint probabilities while discriminative models like logistic regression directly model posterior probabilities.
4) Sigmoid and softmax functions are used to transform linear outputs into probabilities for binary and multiclass classification respectively.
The document discusses properties of determinants:
1. Determinants are functions that assign a scalar value to a square matrix based on properties like how row operations affect the value.
2. Important properties include that the determinant of the identity matrix is 1, and that row operations like scaling a row or exchanging rows affect the determinant in predictable ways.
3. The determinant of a matrix product is equal to the product of the individual determinants, and the determinant of the inverse of an invertible matrix is the inverse of the determinant.
- Estimation theory involves using observed data to determine unknown parameters of a system. This includes problems like estimating locations/velocities from radar signals or inferring transmitted signals from received noisy data.
- Estimation includes parametric estimation, which assumes a model and estimates parameters like mean/variance, and non-parametric estimation, which directly estimates probability densities without assuming a model.
- An estimator is a rule for guessing the value of an unknown parameter based on observed data. Good estimators are unbiased, have low variance, are consistent as more data is observed, and have minimum mean squared error. The minimum variance unbiased estimator is preferred.
The maximum likelihood estimator (MLE) is an estimator that maximizes the likelihood function. For a parameter θ, the MLE θˆMLE is the value of θ that maximizes the likelihood function f(x|θ) given the observed data x. The MLE satisfies the likelihood equation ∂L(θ)/∂θ = 0, where L is the log-likelihood function. For multiple parameters, the MLE satisfies a set of likelihood equations. The MLE is asymptotically unbiased, efficient, and consistent under certain conditions.
This document discusses partial differential equations (PDEs) and methods for solving them. It covers:
1. Classification of first and second order PDEs as elliptic, hyperbolic, or parabolic based on characteristics.
2. The method of separation of variables to solve PDEs by breaking them into ordinary differential equations.
3. Solutions to the one-dimensional wave equation and heat equation under various assumptions about material properties.
This document discusses differential equations and their solutions. It defines differential equations as equations involving derivatives. It notes that solutions can be general, containing an arbitrary constant, or particular, containing an initial value. Examples are given of separating variables and integrating to find the general solution to first order differential equations.
Logarithms relate an input value to the power needed to raise a base to produce that output value. Logarithmic scales are used to measure sound because they match how humans perceive changes in loudness. The key properties of logarithms are:
1) Logarithmic functions are inverses of exponential functions.
2) When working with logarithms or exponents, it helps to rewrite the problem in the other form.
3) For logarithmic equations, setting the arguments equal is valid if the bases are the same.
The document discusses linear and literal equations and inequalities. It defines linear equations as equations with variables of exponent 1 that can be solved by isolating the variable. Literal equations contain two or more variables and specify which variable to solve for. Inequalities are solved like equations but the inequality symbol changes if multiplying or dividing by a negative number. Examples are provided for solving each type.
This document discusses techniques for solving first order differential equations, including separation of variables and exact equations. It begins by introducing separation of variables as a method to solve equations of the form dy/dx = f(x)g(y) by dividing through and integrating both sides separately. Examples are provided. The document then discusses exact equations, which can be solved directly by integrating both sides. It provides examples and explains how the solution consists of a definite part satisfying the equation and an indefinite part satisfying a related equation with zero on the right side. The document concludes by explaining how to recognize exact equations based on their form.
In the implementation of algorithm we are evaluating the indexes of 1-D NN to the alpha by which we will get the relevant recurrence coefficient value in discrete orthogonal polynomial. In mathematics, a recurrence coefficient is an equation that recursively defines a sequence, once one or more initial terms are given and each further term of the sequence is defined as a function of the preceding terms.
We looked at the data. Here’s a breakdown of some key statistics about the nation’s incoming presidents’ addresses, how long they spoke, how well, and more.
The document discusses how startup entrepreneurs think and operate. It notes that startups like Airbnb and Uber were started due to identifying shortages or problems. It emphasizes that startups focus on providing customer benefit, eliminating waste, and creating value. It also highlights that startups operate with speed, embracing failure fast and pivoting quickly, with transparency and by breaking rules. Startups succeed by moving rapidly, with minimal processes and instead prioritizing speed above all else.
1) The document discusses fractions and rational numbers. It notes that US students struggle with fraction arithmetic compared to international peers and provides an example problem where over 42% of US students chose the wrong answer.
2) It advocates understanding fractions as sums of unit fractions (e.g. 2/7 = 1/4 + 1/28) in the same way place value is taught for decimals. This conceptual understanding enables computational fluency with fractions.
3) It distinguishes between fractions, which are written symbols, and rational numbers, which depend on value not notation. A rational number can always be expressed as a quotient of two integers.
This document provides an overview of key concepts in polynomial functions. It defines polynomials as functions where the exponents are positive whole numbers. It explains standard form, factored form, degrees, leading coefficients, roots, ends, and graphing behaviors. It also covers dividing polynomials using long division and synthetic division, and defines the remainder and factor theorems. Specifically, the remainder theorem states the remainder of dividing a polynomial f(x) by (x-k) is r=f(k), while the factor theorem says a polynomial f(x) has a factor (x-k) if and only if f(k)=0.
Exact & non exact differential equationAlaminMdBabu
This document discusses exact and non-exact differential equations. It states that an exact differential equation can be obtained directly by differentiation of its solution, without any other operations. It provides the working rule to check if a differential equation is exact by seeing if the partial derivatives are equal. For non-exact equations, an integrating factor can be found using four different rules depending on the form of the differential equation.
This document provides an overview of Chapter 1 from the textbook "Differential Equations & Linear Algebra" which covers first-order ordinary differential equations. It defines differential equations and their order, provides examples of common types of differential equations and mathematical models, and explains concepts like general/particular solutions and initial value problems. The chapter then covers methods for solving first-order differential equations, including those that are separable, linear, or may require a substitution to transform into a separable or linear equation like the homogeneous or Bernoulli equations. Suggested practice problems are marked for exam inspiration.
1) Linear and nonlinear first order differential equations differ in their theory of existence and uniqueness of solutions, properties of general solutions, and whether solutions can be expressed explicitly or only implicitly.
2) For linear equations, solutions are guaranteed to exist and be unique based on continuity of coefficients, and general solutions involving arbitrary constants can represent all solutions.
3) Nonlinear equations may only have local existence and uniqueness depending on initial conditions, general solutions may not encompass all solutions, and solutions are often only implicitly defined requiring numerical methods.
Oscillation and Convengence Properties of Second Order Nonlinear Neutral Dela...inventionjournals
In this paper, we consider the second order nonlinear neutral delay difference equations of the form We establish sufficient conditions which ensures that every solution of is either oscillatory or tends to zero as . We also gives examples to illustrate our results
Formal expansion method for solving an electrical circuit modelTELKOMNIKA JOURNAL
We investigate the validity of the formal expansion method for solving a second order ordinary differential equation raised from an electrical circuit problem. The formal expansion method approximates the exact solution using a series of solutions. An approximate formal expansion solution is a truncated version of this series. In this paper, we confirm using simulations that the approximate formal expansion solution is valid for a specific interval of domain of the free variable. The accuracy of the formal expansion approximation is guaranteed on the time-scale 1.
This document discusses solving systems of linear equations with two or three variables. It explains that a system can have one solution, infinitely many solutions, or no solution. For two variables, the solutions are where lines intersect (one solution), coincide (infinitely many), or are parallel (no solution). For three variables, the solutions are where planes intersect (one point), lie on a line (infinitely many), or do not intersect (no solution). It demonstrates solving systems using substitution, elimination, and matrix methods, and discusses cases where a system has infinitely many or no solutions.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This chapter discusses classification methods including linear discriminant functions and probabilistic generative and discriminative models. It covers linear decision boundaries, perceptrons, Fisher's linear discriminant, logistic regression, and the use of sigmoid and softmax activation functions. The key points are:
1) Classification involves dividing the input space into decision regions using linear or nonlinear boundaries.
2) Perceptrons and Fisher's linear discriminant find linear decision boundaries by updating weights to minimize misclassification.
3) Generative models like naive Bayes estimate joint probabilities while discriminative models like logistic regression directly model posterior probabilities.
4) Sigmoid and softmax functions are used to transform linear outputs into probabilities for binary and multiclass classification respectively.
The document discusses properties of determinants:
1. Determinants are functions that assign a scalar value to a square matrix based on properties like how row operations affect the value.
2. Important properties include that the determinant of the identity matrix is 1, and that row operations like scaling a row or exchanging rows affect the determinant in predictable ways.
3. The determinant of a matrix product is equal to the product of the individual determinants, and the determinant of the inverse of an invertible matrix is the inverse of the determinant.
- Estimation theory involves using observed data to determine unknown parameters of a system. This includes problems like estimating locations/velocities from radar signals or inferring transmitted signals from received noisy data.
- Estimation includes parametric estimation, which assumes a model and estimates parameters like mean/variance, and non-parametric estimation, which directly estimates probability densities without assuming a model.
- An estimator is a rule for guessing the value of an unknown parameter based on observed data. Good estimators are unbiased, have low variance, are consistent as more data is observed, and have minimum mean squared error. The minimum variance unbiased estimator is preferred.
The maximum likelihood estimator (MLE) is an estimator that maximizes the likelihood function. For a parameter θ, the MLE θˆMLE is the value of θ that maximizes the likelihood function f(x|θ) given the observed data x. The MLE satisfies the likelihood equation ∂L(θ)/∂θ = 0, where L is the log-likelihood function. For multiple parameters, the MLE satisfies a set of likelihood equations. The MLE is asymptotically unbiased, efficient, and consistent under certain conditions.
This document discusses partial differential equations (PDEs) and methods for solving them. It covers:
1. Classification of first and second order PDEs as elliptic, hyperbolic, or parabolic based on characteristics.
2. The method of separation of variables to solve PDEs by breaking them into ordinary differential equations.
3. Solutions to the one-dimensional wave equation and heat equation under various assumptions about material properties.
This document discusses differential equations and their solutions. It defines differential equations as equations involving derivatives. It notes that solutions can be general, containing an arbitrary constant, or particular, containing an initial value. Examples are given of separating variables and integrating to find the general solution to first order differential equations.
Logarithms relate an input value to the power needed to raise a base to produce that output value. Logarithmic scales are used to measure sound because they match how humans perceive changes in loudness. The key properties of logarithms are:
1) Logarithmic functions are inverses of exponential functions.
2) When working with logarithms or exponents, it helps to rewrite the problem in the other form.
3) For logarithmic equations, setting the arguments equal is valid if the bases are the same.
The document discusses linear and literal equations and inequalities. It defines linear equations as equations with variables of exponent 1 that can be solved by isolating the variable. Literal equations contain two or more variables and specify which variable to solve for. Inequalities are solved like equations but the inequality symbol changes if multiplying or dividing by a negative number. Examples are provided for solving each type.
This document discusses techniques for solving first order differential equations, including separation of variables and exact equations. It begins by introducing separation of variables as a method to solve equations of the form dy/dx = f(x)g(y) by dividing through and integrating both sides separately. Examples are provided. The document then discusses exact equations, which can be solved directly by integrating both sides. It provides examples and explains how the solution consists of a definite part satisfying the equation and an indefinite part satisfying a related equation with zero on the right side. The document concludes by explaining how to recognize exact equations based on their form.
In the implementation of algorithm we are evaluating the indexes of 1-D NN to the alpha by which we will get the relevant recurrence coefficient value in discrete orthogonal polynomial. In mathematics, a recurrence coefficient is an equation that recursively defines a sequence, once one or more initial terms are given and each further term of the sequence is defined as a function of the preceding terms.
We looked at the data. Here’s a breakdown of some key statistics about the nation’s incoming presidents’ addresses, how long they spoke, how well, and more.
The document discusses how startup entrepreneurs think and operate. It notes that startups like Airbnb and Uber were started due to identifying shortages or problems. It emphasizes that startups focus on providing customer benefit, eliminating waste, and creating value. It also highlights that startups operate with speed, embracing failure fast and pivoting quickly, with transparency and by breaking rules. Startups succeed by moving rapidly, with minimal processes and instead prioritizing speed above all else.
This document discusses how emojis, emoticons, and text speak can be used to teach students. It provides background on the origins of emoticons in 1982 as ways to convey tone and feelings in text communications. It then suggests that with text speak and emojis, students can translate, decode, summarize, play with language, and add emotion to language. A number of websites and apps that can be used for emoji-related activities, lessons, and discussions are also listed.
Artificial intelligence (AI) is everywhere, promising self-driving cars, medical breakthroughs, and new ways of working. But how do you separate hype from reality? How can your company apply AI to solve real business problems?
Here’s what AI learnings your business should keep in mind for 2017.
Study: The Future of VR, AR and Self-Driving CarsLinkedIn
We asked LinkedIn members worldwide about their levels of interest in the latest wave of technology: whether they’re using wearables, and whether they intend to buy self-driving cars and VR headsets as they become available. We asked them too about their attitudes to technology and to the growing role of Artificial Intelligence (AI) in the devices that they use. The answers were fascinating – and in many cases, surprising.
This SlideShare explores the full results of this study, including detailed market-by-market breakdowns of intention levels for each technology – and how attitudes change with age, location and seniority level. If you’re marketing a tech brand – or planning to use VR and wearables to reach a professional audience – then these are insights you won’t want to miss.
Solving systems of equations in 3 variablesJessica Garcia
The document discusses solving systems of linear equations with two or three variables. For two variables, the solutions can be one point (consistent, one solution), all points on a line (consistent, dependent with infinite solutions), or no solution (inconsistent). For three variables, the solutions can be one point, all points on a line, or no solution, which are determined by whether the graphs (planes) intersect at a point, line, or not at all. The document demonstrates using elimination methods to solve sample systems of two and three variable equations algebraically.
The document discusses solving systems of 3 linear equations with 3 variables. It provides steps to set up the equations in standard form, eliminate one variable using two equations, eliminate the same variable from another pair of equations to get a system of 2 equations with 2 variables, solve this system to find the values of two variables, substitute these values into the original third equation to solve for the remaining variable, and check the solution. An example problem demonstrates applying these steps to solve for x, y, and z. The summary notes that the solution may not be unique depending on whether eliminating variables results in a true or false statement.
This document provides an overview of solving nonlinear equations. It discusses solving single nonlinear equations and systems of nonlinear equations. It covers existence and uniqueness of solutions, multiple roots, sensitivity and conditioning. Iterative methods are used to solve nonlinear equations numerically. The bisection method is introduced, which successively halves the interval containing the solution until the desired accuracy is reached. The convergence rate of the bisection method is linear, meaning the error bound is reduced by half each iteration.
A power series is a series of the form Σan(x-x0)n where x0 and an are numbers. A power series converges for x=c if the limit of the partial sums exists and is finite. The radius of convergence of a power series can be found using the ratio test, and determines the values of x for which the power series converges. Basic operations like addition, multiplication, differentiation, and index shifts can be performed on power series term-by-term.
This document provides an overview of solving linear equations, formulas, and problem solving techniques. It begins by introducing the basic properties of equality used to solve linear equations, such as distributing terms and adding/subtracting terms to isolate the variable. Examples are provided to demonstrate solving equations with fractions and solving literal equations for a specified variable. The document also discusses identities, contradictions, and using a general formula to solve families of linear equations. It concludes by outlining a problem solving guide to organize the steps of reading, visualizing, and developing an equation model to solve word problems.
The document discusses the natural logarithm function ln(x) and the natural exponential function exp(x). It begins by defining ln(x) as the area under the curve y=1/t from 1 to x, and noting that its derivative is 1/x. It then defines exp(x) as the inverse of ln(x). It is shown that for rational r, exp(r) = er, and this definition is extended to irrational r. The derivative of exp(x) is then shown to be exp(x) itself.
This document provides an overview of solving linear equations, including:
- Slides cover solving linear equations and practice questions
- Includes an algebra cheat sheet and guide for using a graphing calculator
- Defines a linear equation and provides examples of solving linear equations algebraically by isolating the variable
- Shows checking solutions by plugging them back into the original equation
- Provides word problems applying linear equations to real-world scenarios like temperature conversion and asset depreciation
The document discusses linear programming problems and their graphical solutions. It introduces:
- Graphing linear inequalities in two variables by representing the solution set as a half-plane defined by the inequality. Any point on or below the graph line satisfies the inequality.
- Solving linear programming problems with two unknowns using graphical methods by representing the feasible region as the intersection of half-planes defined by the constraints.
- More advanced algebraic methods, like the simplex method, for solving problems with three or more unknowns.
This document provides instruction on solving various types of equations, including quadratic equations, equations containing radicals, and equations that can be reduced to quadratic form. It includes examples of solving equations that are quadratic in form by rewriting them in standard form and substituting a variable, then solving the resulting quadratic equation. It also provides examples of solving equations containing radicals by isolating radicals and raising both sides of the equation to appropriate powers. Students are expected to learn to identify different equation types, select appropriate solution methods, and solve various quadratic and radical equations.
Solving systems of equations in 3 variablesJessica Garcia
1. The solution to a system of linear equations with three variables can take three forms: a unique solution, infinitely many solutions, or no solution.
2. If the graphs of the three planes all intersect at a single point, there is a unique solution. If the planes intersect along a line, there are infinitely many solutions. If the planes do not intersect at all, there is no solution.
3. The method of elimination is used to solve these systems by eliminating variables one at a time until reaching the solution.
The document provides a summary and explanation of practice problems from an AP Calculus exam review session. It outlines solutions to questions on finding derivatives up to the fourth order, using the Fundamental Theorem of Calculus, finding tangent lines and intersections, optimizing an area problem using derivatives, applying the Second Fundamental Theorem of Calculus, and using separation of variables and derivative rules. It reminds students of upcoming exam deadlines and assignments.
Calculus is the study of change and is divided into differential and integral calculus. Differential calculus studies rates of change using derivatives, while integral calculus uses integration to find accumulated change. These concepts build on limits and algebra/geometry. Leibniz developed the notation and principles of calculus in the 1670s. Differential calculus uses derivatives to determine how quantities change, and integral calculus uses integrals and antiderivatives to determine quantities from rates of change. Differential equations relate functions to their derivatives and have general solutions representing families of curves.
1) Solving linear inequalities involves using the same concepts as solving linear equations to isolate the variable, but inequalities have infinite solutions rather than a single solution.
2) Inequality signs (<, >, ≤, ≥) indicate whether values are less than, greater than, less than or equal to, or greater than or equal to another value.
3) Graphing linear inequalities on a number line or coordinate plane involves graphing the corresponding equality and then shading the appropriate side based on the inequality sign and whether it includes the endpoint or not.
Logarithms relate an input value to the power needed to raise a base to produce that output value. Logarithmic scales are used to measure sound because they allow for large ranges of intensity to be represented on a linear scale. The key properties of logarithms are:
1) Logarithmic functions are inverses of exponential functions.
2) When working with logarithms or exponentials, it helps to rewrite the problem in the other form.
3) For logarithmic equations, setting the arguments equal is valid if the bases are the same.
This document summarizes an algorithm writing technique using invariants inspired by Professor Dijkstra's style of proving and deriving algorithms. It explains what invariants and predicates are, and how to use Hoare triples to prove correctness of while loops. An example is provided to derive an algorithm to calculate 2^n using an invariant. Exercises are provided to practice deriving algorithms for various problems like calculating sums, finding maximums, sorting arrays, and more using invariants.
This document provides information about solving absolute value equations and inequalities, as well as quadratic equations. It discusses:
1) To solve absolute value equations, you must divide the equation into two equations by treating the expression inside the absolute value bars as both positive and negative.
2) For inequalities, the direction of the inequality sign must be reversed when multiplying or dividing both sides by a negative number.
3) Quadratic equations can be solved by factoring if possible, or using the quadratic formula. The discriminant determines the number of real roots.
This document discusses solving linear inequalities and absolute value inequalities. It introduces inequality notation such as <, >, ≤, ≥ and explains that solutions to inequalities are intervals of the number line rather than single numbers. When solving inequalities, the direction of the inequality must be reversed when multiplying or dividing by a negative number. Worked examples are provided to demonstrate solving linear inequalities by applying the basic arithmetic rules and reversing the inequality direction with negative coefficients.
This document provides tips for acing the Additional Mathematics (AM) and Elementary Mathematics (EM) exams. It summarizes key statistics on topics that are highly tested, such as differentiation and integration making up 27.8% of the AM exam. It recommends focusing on the 11 chapters that make up 74.6% of the exam. Sample questions are provided for topics like trigonometry, logarithms, linear laws, and matrices. Strategies are outlined for solving different types of questions on these topics.
This chapter discusses continuous latent variable models including principal component analysis (PCA), probabilistic PCA, and factor analysis. PCA finds projections of data that maximize variance or minimize error through eigenvectors of the covariance matrix. Probabilistic PCA places a probabilistic treatment on PCA by modeling the data and latent variables as Gaussian distributions. Factor analysis similarly models the data as a linear combination of latent factors plus noise.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.
Infrastructure Challenges in Scaling RAG with Custom AI modelsZilliz
Building Retrieval-Augmented Generation (RAG) systems with open-source and custom AI models is a complex task. This talk explores the challenges in productionizing RAG systems, including retrieval performance, response synthesis, and evaluation. We’ll discuss how to leverage open-source models like text embeddings, language models, and custom fine-tuned models to enhance RAG performance. Additionally, we’ll cover how BentoML can help orchestrate and scale these AI components efficiently, ensuring seamless deployment and management of RAG systems in the cloud.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
AI 101: An Introduction to the Basics and Impact of Artificial IntelligenceIndexBug
Imagine a world where machines not only perform tasks but also learn, adapt, and make decisions. This is the promise of Artificial Intelligence (AI), a technology that's not just enhancing our lives but revolutionizing entire industries.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
1. If your familiar with linear algebra you'll recall that a transformation is called linear if T(v + w) = T(v) + T(w). So what we are really seeing is that the left hand side of the ODE is a linear transformation on functions, and it is for this reason the equation is called linear.<br />http://www.cliffsnotes.com/study_guide/Linear-Transformations.topicArticleId-19736,articleId-19730.htmlIf you are faced with an IVP that involves a linear differential equation with constant coefficients, you can proceed by the method of undetermined coefficients or variation of parameters and then apply the initial conditions to evaluate the constants. However, what if the nonhomogeneous right-hand term is discontinuous? There exists a method for solving such problems that can also be used to solve less frightening IVP's (that is, ones that do not involve discontinuous terms) and even some equations whose coefficients are not constants. One of the features of this alternative method for solving IVP's is that the values of the parameters are not found after the general solution has been obtained. Instead, the initial conditions are incorporated right into the initial stages of the solution, so when the final step is completed, the arbitrary constants have already been evaluated.<br />http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx Dars & mesal<br />Separable Differential Equations<br />We are now going to start looking at nonlinear first order differential equations. The first type of nonlinear first order differential equations that we will look at is separable differential equations. <br /> <br />A separable differential equation is any differential equation that we can write in the following form.<br />(1)<br /> <br />Note that in order for a differential equation to be separable all the y's in the differential equation must be multiplied by the derivative and all the x's in the differential equation must be on the other side of the equal sign. <br /> <br />Solving separable differential equation is fairly easy. We first rewrite the differential equation as the following<br /> <br /> <br />Then you integrate both sides.<br />(2)<br /> <br />So, after doing the integrations in (2) you will have an implicit solution that you can hopefully solve for the explicit solution, y(x). Note that it won't always be possible to solve for an explicit solution.<br /> <br />Recall from the Definitions section that an implicit solution is a solution that is not in the form while an explicit solution has been written in that form. <br /> <br />We will also have to worry about the interval of validity for many of these solutions. Recall that the interval of validity was the range of the independent variable, x in this case, on which the solution is valid. In other words, we need to avoid division by zero, complex numbers, logarithms of negative numbers or zero, etc. Most of the solutions that we will get from separable differential equations will not be valid for all values of x.<br /> <br />Let’s start things off with a fairly simple example so we can see the process without getting lost in details of the other issues that often arise with these problems.<br /> <br />Example 1 Solve the following differential equation and determine the interval of validity for the solution. SolutionIt is clear, hopefully, that this differential equation is separable. So, let’s separate the differential equation and integrate both sides. As with the linear first order officially we will pick up a constant of integration on both sides from the integrals on each side of the equal sign. The two can be moved to the same side an absorbed into each other. We will use the convention that puts the single constant on the side with the x’s. So, we now have an implicit solution. This solution is easy enough to get an explicit solution, however before getting that it is usually easier to find the value of the constant at this point. So apply the initial condition and find the value of c. Plug this into the general solution and then solve to get an explicit solution. Now, as far as solutions go we’ve got the solution. We do need to start worrying about intervals of validity however. Recall that there are two conditions that define an interval of validity. First, it must be a continuous interval with no breaks or holes in it. Second it must contain the value of the independent variable in the initial condition, x = 1 in this case. So, for our case we’ve got to avoid two values of x. Namely, since these will give us division by zero. This gives us three possible intervals of validity. However, only one of these will contain the value of x from the initial condition and so we can see that must be the interval of validity for this solution. Here is a graph of the solution. Note that this does not say that either of the other two intervals listed above can’t be the interval of validity for any solution. With the proper initial condition either of these could have been the interval of validity. We’ll leave it to you to verify the details of the following claims. If we use an initial condition of we will get exactly the same solution however in this case the interval of validity would be the first one. Likewise, if we use as the initial condition we again get exactly the same solution and in this case the third interval becomes the interval of validity.So, simply changing the initial condition a little can give any of the possible intervals.<br /> <br />Example 2 Solve the following IVP and find the interval of validity for the solution. Solution This differential equation is clearly separable, so let's put it in the proper form and then integrate both sides. We now have our implicit solution, so as with the first example let’s apply the initial condition at this point to determine the value of c. The implicit solution is then We now need to find the explicit solution. This is actually easier than it might look and you already know how to do it. First we need to rewrite the solution a little To solve this all we need to recognize is that this is quadratic in y and so we can use the quadratic formula to solve it. However, unlike quadratics you are used to, at least some of the “constants” will not actually be constant, but will in fact involve x’s. So, upon using the quadratic formula on this we get. Next, notice that we can factor a 4 out from under the square root (it will come out as a 2…) and then simplify a little. We are almost there. Notice that we’ve actually got two solutions here (the “ ”) and we only want a single solution. In fact, only one of the signs can be correct. So, to figure out which one is correct we can reapply the initial condition to this. Only one of the signs will give the correct value so we can use this to figure out which one of the signs is correct. Plugging x = 1 into the solution gives. In this case it looks like the “+” is the correct sign for our solution. Note that it is completely possible that the “” could be the solution so don’t always expect it to be one or the other. The explicit solution for our differential equation is. To finish the example out we need to determine the interval of validity for the solution. If we were to put a large negative value of x in the solution we would end up with complex values in our solution and we want to avoid complex numbers in our solutions here. So, we will need to determine which values of x will give real solutions. To do this we will need to solve the following inequality. In other words, we need to make sure that the quantity under the radical stays positive. Using a computer algebra system like Maple or Mathematica we see that the left side is zero at x = 3.36523 as well as two complex values, but we can ignore complex values for interval of validity computations. Finally a graph of the quantity under the radical is shown below. So, in order to get real solutions we will need to require because this is the range of x’s for which the quantity is positive. Notice as well that this interval also contains the value of x that is in the initial condition as it should. Therefore, the interval of validity of the solution is . Here is graph of the solution.<br /> <br />Example 3 Solve the following IVP and find the interval of validity of the solution. SolutionFirst separate and then integrate both sides. Apply the initial condition to get the value of c. The implicit solution is then, Now let’s solve for y(x). Reapplying the initial condition shows us that the “” is the correct sign. The explicit solution is then, Let’s get the interval of validity. That’s easier than it might look for this problem. First, since the “inner” root will not be a problem. Therefore all we need to worry about is division by zero and negatives under the “outer” root. We can take care of both be requiring Note that we were able to square both sides of the inequality because both sides of the inequality are guaranteed to be positive in this case. Finally solving for x we see that the only possible range of x’s that will not give division by zero or square roots of negative numbers will be, and nicely enough this also contains the initial condition x=0. This interval is therefore our interval of validity. Here is a graph of the solution.<br /> <br />Example 4 Solve the following IVP and find the interval of validity of the solution. Solution This differential equation is easy enough to separate, so let's do that and then integrate both sides. Applying the initial condition gives This then gives an implicit solution of. We can easily find the explicit solution to this differential equation by simply taking the natural log of both sides. Finding the interval of validity is the last step that we need to take. Recall that we can't plug negative values or zero into a logarithm, so we need to solve the following inequality The quadratic will be zero at the two points . A graph of the quadratic (shown below) shows that there are in fact two intervals in which we will get positive values of the polynomial and hence can be possible intervals of validity.So, possible intervals of validity are From the graph of the quadratic we can see that the second one contains x = 5, the value of the independent variable from the initial condition. Therefore the interval of validity for this solution is. Here is a graph of the solution.<br /> <br />Example 5 Solve the following IVP and find the interval of validity for the solution. SolutionThis is actually a fairly simple differential equation to solve. I’m doing this one mostly because of the interval of validity. So, get things separated out and then integrate. Now, apply the initial condition to find c. So, the implicit solution is then, Solving for r gets us our explicit solution. Now, there are two problems for our solution here. First we need to avoid θ = 0 because of the natural log. Notice that because of the absolute value on the θ we don’t need to worry about θ being negative. We will also need to avoid division by zero. In other words, we need to avoid the following points. So, these three points break the number line up into four portions, each of which could be an interval of validity. The interval that will be the actual interval of validity is the one that contains θ = 1. Therefore, the interval of validity is . Here is a graph of the solution.<br /> <br />Example 6 Solve the following IVP. Solution This problem will require a little work to get it separated and in a form that we can integrate, so let's do that first. Now, with a little integration by parts on both sides we can get an implicit solution. Applying the initial condition gives. Therefore, the implicit solution is. It is not possible to find an explicit solution for this problem and so we will have to leave the solution in its implicit form. Finding intervals of validity from implicit solutions can often be very difficult so we will also not bother with that for this problem.<br />