This document discusses series solutions and special functions in engineering mathematics. It covers:
1) Finding series solutions to second order differential equations with variable coefficients by expressing the solution as a power series.
2) Bessel equations and their series solutions, including the general solutions for non-integer and integer orders of the Bessel equation.
3) Properties of Bessel functions, Legendre polynomials, and their generating functions. This includes orthogonality relations and recurrence relations.
Bessel functions are solutions to Bessel's differential equation and describe oscillations that arise in many physical systems. Friedrich Bessel first systematically analyzed solutions to this equation in 1824, which became known as Bessel functions. There are Bessel functions of the first kind (Jp(x)) and second kind (Yp(x)). Jp(x) is bounded at x=0 while Yp(x) is unbounded, making them linearly independent solutions for the general solution. The gamma function was developed to define Bessel functions for all real values of p.
This document discusses Bessel's equation and its solutions. It begins by defining Bessel's equation of order ν. It then examines specific cases for ν = 0, 1/2, and 1. For each case it derives the recurrence relation and finds the first solution, which is a Bessel function of the first kind for that order. It also discusses the Bessel function of the second kind for order zero. The document provides graphs and discusses approximations of Bessel functions for large values of x.
1) Friedrich Wilhelm Bessel studied disturbances in planetary motion in 1824, which led him to analyze solutions to the Bessel equation.
2) The Bessel equation is a second-order linear ordinary differential equation whose solutions are called Bessel functions.
3) Bessel functions have many applications in physics and engineering, particularly in problems involving cylindrical and spherical coordinates.
Power Series - Legendre Polynomial - Bessel's EquationArijitDhali
The presentation shows types of equations inside every topic along with its general form, generating formula, and other equations like recursion, frobenius, rodrigues etc for calculus. Its an overall explanation in a brief. You are at correct link to get your work done out of this in your engineering maths.
In this presentation we can get to know the meaning of basic discrete distribution for bivariate. There are also discussions regarding the topic along with marginal tables. Also there are certain illustrative example for the ease of understanding. Overall it is a great presentation for the junior engineers aiming in their course.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
- 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 introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
Bessel functions are solutions to Bessel's differential equation and describe oscillations that arise in many physical systems. Friedrich Bessel first systematically analyzed solutions to this equation in 1824, which became known as Bessel functions. There are Bessel functions of the first kind (Jp(x)) and second kind (Yp(x)). Jp(x) is bounded at x=0 while Yp(x) is unbounded, making them linearly independent solutions for the general solution. The gamma function was developed to define Bessel functions for all real values of p.
This document discusses Bessel's equation and its solutions. It begins by defining Bessel's equation of order ν. It then examines specific cases for ν = 0, 1/2, and 1. For each case it derives the recurrence relation and finds the first solution, which is a Bessel function of the first kind for that order. It also discusses the Bessel function of the second kind for order zero. The document provides graphs and discusses approximations of Bessel functions for large values of x.
1) Friedrich Wilhelm Bessel studied disturbances in planetary motion in 1824, which led him to analyze solutions to the Bessel equation.
2) The Bessel equation is a second-order linear ordinary differential equation whose solutions are called Bessel functions.
3) Bessel functions have many applications in physics and engineering, particularly in problems involving cylindrical and spherical coordinates.
Power Series - Legendre Polynomial - Bessel's EquationArijitDhali
The presentation shows types of equations inside every topic along with its general form, generating formula, and other equations like recursion, frobenius, rodrigues etc for calculus. Its an overall explanation in a brief. You are at correct link to get your work done out of this in your engineering maths.
In this presentation we can get to know the meaning of basic discrete distribution for bivariate. There are also discussions regarding the topic along with marginal tables. Also there are certain illustrative example for the ease of understanding. Overall it is a great presentation for the junior engineers aiming in their course.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
- 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 introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
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.
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.
The document provides information about solving systems of linear and nonlinear equations:
- It discusses the three types of solutions to systems of linear equations in two and three variables - one solution, no solution, and infinitely many solutions. Methods for solving include substitution, elimination, and graphing.
- Systems of nonlinear equations like quadratics are also introduced.
- Several examples of application problems involving systems of equations with two or three variables are presented and solved.
This document lists 19 different inequalities involving sums, products, and ratios of positive real numbers. Some of the key inequalities presented include:
1) The General AM-GM Inequality, which states the arithmetic mean of positive numbers is greater than or equal to their geometric mean.
2) Hölder's Inequality, which provides an upper bound for the product of sums of component-wise multiplications of positive real numbers.
3) Minkowski's Inequality, a generalization of the triangle inequality for Lp spaces that gives an upper bound for sums of pth powers of vectors.
4) Jensen's Inequality, concerning convex/concave functions applied to averages, and how averages relate to
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.
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.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
This document discusses orthogonal trajectories and provides examples of finding orthogonal trajectories for different families of curves. It begins by defining orthogonal trajectories as curves that intersect each other at right angles. It then provides a method for finding the differential equation that describes the orthogonal trajectories for a given family of curves. Several examples are worked out, such as finding the orthogonal trajectories of the family of parabolas with equation y = x^2. Applications to equipotential lines and electric fields and electromagnetic waves are also mentioned.
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.
Differential equation and Laplace transformsujathavvv
- Differential equations relate an unknown function and its derivatives, and are classified as ordinary (ODE) or partial (PDE) depending on the number of independent variables. Higher order equations can be reduced to systems of first order equations.
- Exact differential equations can be written as the total differential of a function, while non-exact equations may require an integrating factor to convert them to exact form.
- The general solution to a non-homogeneous second order differential equation is the sum of the complementary function (solution to the homogeneous equation) and a particular integral similar to the non-homogeneous term.
Differential equation and Laplace Transformsujathavvv
The document discusses several topics in mathematics including:
1. Differential equations in the form of Ay" + By′ + cy = g(t) and their homogeneous solutions.
2. Classifications of integrals including singular integrals which are found by eliminating arbitrary constants from an integral equation.
3. Solving Lagrange's linear equations by writing subsidiary equations and finding integrals to get the general integral solution in the form φ(u,v)=0.
4. The Laplace transform of a function f(t), denoted by L{f(t)}, which is defined as the integral of f(t)e^-st from 0 to infinity if it exists. Some properties of the Laplace transform are
The document presents a multi-step problem to determine the area bounded by a circle, the x-axis, y-axis, and the line y=12. It provides the equation of the circle and guides the reader through simplifying systems of equations to find values for variables in the circle equation. These are used to graph the circle and identify the bounded area as a rectangle minus one quarter of the circle area, calculated as 66.9027 square units.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
The document discusses the elimination method for solving simultaneous equations. It involves adding or subtracting two equations to eliminate one of the variables. This leaves an equation with only one variable that can then be solved for. Examples are provided to demonstrate how to eliminate variables by adding or subtracting equations. It notes that sometimes equations need to be multiplied by a number first to get equal terms that can cancel out before elimination can occur. The goal is to reduce the simultaneous equations down to one equation with one variable that can then be solved.
1. There are two characteristics that dictate an exponential behavior: the base x must be a decimal with an absolute value less than 1, and it must be a negative number.
2. The document provides examples and explanations for why certain exponential expressions with a negative decimal base will always be positive or negative.
3. It evaluates various inequalities involving exponential expressions with a negative decimal base to determine relationships between the expressions.
1. The angles labeled (2a)° and (5a + 5)° are supplementary and add up to 180°. Solving for a gives a = 25. Similarly, the angles labeled (4b +10)° and (2b – 10)° are supplementary and add up to 180°. Solving for b gives b = 30. Therefore, a + b = 25 + 30 = 55.
2. Two parallel lines intersected by a transversal form eight angles. The acute angles are equal and the obtuse angles are equal. The acute angles are supplementary to the obtuse angles. Solving the equation relating the angles gives the answer.
3. The relationship between angles formed when
The document discusses ordinary differential equations, including exponential growth/decay models, separation of variables, numerical and hybrid numerical-symbolic solving techniques, orthogonal curves, Newton's law of heating and cooling, and medical modeling examples. Specific examples are provided to illustrate concepts like families of solutions, implicit solutions, direction fields, and determining parameter values from initial conditions.
This document discusses generalizing the concept of regularity for semigroups in two ways: higher regularity and higher arity (polyadic semigroups).
For binary semigroups, higher n-regularity is defined such that each element has multiple inverse elements rather than a single inverse. However, for binary semigroups this reduces to ordinary regularity. For polyadic semigroups, several definitions of regularity and higher regularity are introduced to account for the higher arity operations. Idempotents and identities are also generalized for polyadic semigroups. It is shown that the definitions of regularity for polyadic semigroups cannot be reduced in the same
First order non-linear partial differential equation & its applicationsJayanshu Gundaniya
There are five types of methods for solving first order non-linear partial differential equations:
I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.
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.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
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.
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.
The document provides information about solving systems of linear and nonlinear equations:
- It discusses the three types of solutions to systems of linear equations in two and three variables - one solution, no solution, and infinitely many solutions. Methods for solving include substitution, elimination, and graphing.
- Systems of nonlinear equations like quadratics are also introduced.
- Several examples of application problems involving systems of equations with two or three variables are presented and solved.
This document lists 19 different inequalities involving sums, products, and ratios of positive real numbers. Some of the key inequalities presented include:
1) The General AM-GM Inequality, which states the arithmetic mean of positive numbers is greater than or equal to their geometric mean.
2) Hölder's Inequality, which provides an upper bound for the product of sums of component-wise multiplications of positive real numbers.
3) Minkowski's Inequality, a generalization of the triangle inequality for Lp spaces that gives an upper bound for sums of pth powers of vectors.
4) Jensen's Inequality, concerning convex/concave functions applied to averages, and how averages relate to
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.
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.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
This document discusses orthogonal trajectories and provides examples of finding orthogonal trajectories for different families of curves. It begins by defining orthogonal trajectories as curves that intersect each other at right angles. It then provides a method for finding the differential equation that describes the orthogonal trajectories for a given family of curves. Several examples are worked out, such as finding the orthogonal trajectories of the family of parabolas with equation y = x^2. Applications to equipotential lines and electric fields and electromagnetic waves are also mentioned.
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.
Differential equation and Laplace transformsujathavvv
- Differential equations relate an unknown function and its derivatives, and are classified as ordinary (ODE) or partial (PDE) depending on the number of independent variables. Higher order equations can be reduced to systems of first order equations.
- Exact differential equations can be written as the total differential of a function, while non-exact equations may require an integrating factor to convert them to exact form.
- The general solution to a non-homogeneous second order differential equation is the sum of the complementary function (solution to the homogeneous equation) and a particular integral similar to the non-homogeneous term.
Differential equation and Laplace Transformsujathavvv
The document discusses several topics in mathematics including:
1. Differential equations in the form of Ay" + By′ + cy = g(t) and their homogeneous solutions.
2. Classifications of integrals including singular integrals which are found by eliminating arbitrary constants from an integral equation.
3. Solving Lagrange's linear equations by writing subsidiary equations and finding integrals to get the general integral solution in the form φ(u,v)=0.
4. The Laplace transform of a function f(t), denoted by L{f(t)}, which is defined as the integral of f(t)e^-st from 0 to infinity if it exists. Some properties of the Laplace transform are
The document presents a multi-step problem to determine the area bounded by a circle, the x-axis, y-axis, and the line y=12. It provides the equation of the circle and guides the reader through simplifying systems of equations to find values for variables in the circle equation. These are used to graph the circle and identify the bounded area as a rectangle minus one quarter of the circle area, calculated as 66.9027 square units.
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
The document discusses the elimination method for solving simultaneous equations. It involves adding or subtracting two equations to eliminate one of the variables. This leaves an equation with only one variable that can then be solved for. Examples are provided to demonstrate how to eliminate variables by adding or subtracting equations. It notes that sometimes equations need to be multiplied by a number first to get equal terms that can cancel out before elimination can occur. The goal is to reduce the simultaneous equations down to one equation with one variable that can then be solved.
1. There are two characteristics that dictate an exponential behavior: the base x must be a decimal with an absolute value less than 1, and it must be a negative number.
2. The document provides examples and explanations for why certain exponential expressions with a negative decimal base will always be positive or negative.
3. It evaluates various inequalities involving exponential expressions with a negative decimal base to determine relationships between the expressions.
1. The angles labeled (2a)° and (5a + 5)° are supplementary and add up to 180°. Solving for a gives a = 25. Similarly, the angles labeled (4b +10)° and (2b – 10)° are supplementary and add up to 180°. Solving for b gives b = 30. Therefore, a + b = 25 + 30 = 55.
2. Two parallel lines intersected by a transversal form eight angles. The acute angles are equal and the obtuse angles are equal. The acute angles are supplementary to the obtuse angles. Solving the equation relating the angles gives the answer.
3. The relationship between angles formed when
The document discusses ordinary differential equations, including exponential growth/decay models, separation of variables, numerical and hybrid numerical-symbolic solving techniques, orthogonal curves, Newton's law of heating and cooling, and medical modeling examples. Specific examples are provided to illustrate concepts like families of solutions, implicit solutions, direction fields, and determining parameter values from initial conditions.
This document discusses generalizing the concept of regularity for semigroups in two ways: higher regularity and higher arity (polyadic semigroups).
For binary semigroups, higher n-regularity is defined such that each element has multiple inverse elements rather than a single inverse. However, for binary semigroups this reduces to ordinary regularity. For polyadic semigroups, several definitions of regularity and higher regularity are introduced to account for the higher arity operations. Idempotents and identities are also generalized for polyadic semigroups. It is shown that the definitions of regularity for polyadic semigroups cannot be reduced in the same
First order non-linear partial differential equation & its applicationsJayanshu Gundaniya
There are five types of methods for solving first order non-linear partial differential equations:
I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.
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.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
This document discusses methods for finding series solutions to differential equations, namely the power series method and Frobenius method. It first introduces the concepts of ordinary and singular points. Then it explains how to apply the power series method to find a series solution for ordinary points by assuming the solution is a power series and equating coefficients. Next, it describes how the Frobenius method extends the power series method to find solutions near singular points by using the indicial equation and recurrence relations. It outlines the three cases for the indicial roots that determine the final form of the solutions.
Mathematical formulation of inverse scattering and korteweg de vries equationAlexander Decker
This document summarizes the mathematical formulation of inverse scattering and the Korteweg-de Vries (KdV) equation. It begins by defining inverse scattering as determining solutions to differential equations based on known asymptotic solutions, specifically by solving the Marchenko equation. It then discusses how the KdV equation describes shallow water waves and solitons, and how the inverse scattering transform method can be used to determine soliton solutions from arbitrary initial conditions. The document outlines the procedure, including deriving the scattering data from an initial potential function and using its time evolution to reconstruct solutions to the KdV equation at later times. It provides examples using reflectionless potentials, specifically obtaining the single-soliton solution from an initial sech^2
An Algebraic Foundation For Factoring Linear Boundary ProblemsLisa Riley
This document presents an algebraic framework for factoring linear boundary problems. It defines an abstract boundary problem as a pair consisting of a surjective linear map and an orthogonally closed subspace of the dual space. It discusses composing boundary problems by composing the linear maps and combining the boundary conditions. The composition of two regular boundary problems corresponds to composing their Green's operators in reverse order. It also characterizes all factorizations of a boundary problem into two smaller problems and discusses how this allows factoring higher-order problems into lower-order subproblems.
This document provides an overview of modular arithmetic and its applications. It begins with definitions of modular equivalence and congruence classes. Applications discussed include hashing to store records efficiently, generating pseudorandom numbers, and calculating ISBN numbers and Easter dates. The Chinese Remainder Theorem and solving linear congruences are also covered. The document concludes with a method for performing arithmetic on large integers using modular arithmetic.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
Differential equation and Laplace transformMohanamalar8
- Differential equations relate an unknown function and its derivatives, and are classified as ordinary (ODE) or partial (PDE) depending on the number of independent variables. Higher order equations can be reduced to systems of first order equations.
- Exact differential equations can be written as the total differential of a function, while non-exact equations may require an integrating factor to convert them to exact form.
- The general solution to a non-homogeneous second order differential equation is the sum of the complementary function (solution to the homogeneous equation) and a particular integral similar to the non-homogeneous term.
The document discusses various topics related to differential equations including:
1. Linear differential equations of nth order with constant coefficients and the characteristic equation. The general solution depends on whether the roots of the characteristic equation are real/complex, distinct/multiple.
2. Finding the complementary function and particular integral to get the general solution for a second order differential equation with constant coefficients.
3. Solving simultaneous linear differential equations using the D-operator method and Laplace transform method.
4. Changing dependent and independent variables to solve second order differential equations numerically using an approximation technique similar to the trapezoid rule.
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
{\displaystyle z=f^{-1}(ze^{z})=W(ze^{z})} z=f^{-1}(ze^{z})=W(ze^{z})
By substituting {\displaystyle z'=ze^{z}} z'=ze^{z} in the above equation, we get the defining equation for the W function (and for the W relation in general):
{\displaystyle z'=W(z')e^{W(z')}} z'=W(z')e^{W(z')}
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
We make use of the conformal compactification of Minkowski spacetime M# to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime [M#]−1 obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying M# with the projective light cone in (4+2)-dimensional spacetime, we write two independent conformal-invariant functionals of the 6-dimensional Maxwellian field strength tensors - one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.
This document discusses higher order differential equations and their applications. It introduces second order homogeneous differential equations and their solutions based on the nature of the roots. Non-homogeneous differential equations are also discussed, along with their general solution being the sum of the solution to the homogeneous equation and a particular solution. Methods for solving non-homogeneous equations are presented, including undetermined coefficients and reduction of order. Applications to problems in various domains like physics, engineering, and circuits are also outlined.
This document discusses Cauchy-Euler differential equations. It begins by defining Cauchy-Euler equations as linear differential equations where the degree of the term matches the order of differentiation. It then provides methods for solving Cauchy-Euler equations of different orders. For first order equations, it shows how to separate variables. For second order equations, it substitutes a power function solution and sets the result equal to zero. For a third order example, it finds the power function exponents that satisfy the equation. The document concludes by discussing the importance and applications of Cauchy-Euler equations.
This document contains solutions to several problems involving vector calculus and partial differential equations.
For problem 1, key points include: deriving an identity involving curl and dot products; showing that curl is self-adjoint under certain boundary conditions where the vector field is parallel to the boundary normal; and explaining how Maxwell's equations with these boundary conditions would produce oscillating electromagnetic wave solutions.
Problem 2 involves solving the eigenproblem for the Laplacian in an annular region using separation of variables. Continuity conditions at the inner and outer radii lead to a transcendental equation determining the eigenvalues.
Problem 3 examines eigenproblems for the Laplacian and curl operators, showing they are self-adjoint and obtaining matrix and finite difference
This document summarizes the use of the Ritz method to approximate the critical frequencies of a tapered hollow beam. It begins by introducing the governing equations and describing the uniform beam solution. It then outlines the Ritz method, which uses the uniform beam eigenfunctions as a basis to approximate the tapered beam solution. The method is applied numerically to predict the first three critical frequencies of the tapered beam, which are found to match well with finite element analysis results. The Ritz method is concluded to be an effective way to approximate critical frequencies for more complex beam geometries.
This document discusses solutions to higher order differential equations with constant coefficients. It begins by introducing homogeneous linear equations of order two or higher of the form y'' + ay' + by + c = 0. It then presents the method of solving such equations by finding the roots of the characteristic or auxiliary equation. Depending on whether the roots are real/distinct, real/equal, or complex conjugates, the general solution will take different forms involving exponential or trigonometric functions. Several examples are worked through. The document also discusses solving higher order differential equations and presents solutions to sample third and fourth order equations.
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The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
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2. Series Solution and Special Functions ...................................................................................... 2
Series solution of second order ordinary differential equations with variable coefficient ..................................2
Bessel equations and their series solutions ......................................................................................5
Properties of Bessel function ................................................................................................. 11
Properties of Legendre polynomials ............................................................................................... 15
Generating function............................................................................................................................................15
Rodrigues’ Formula.............................................................................................................................................16
Series Solution and Special Functions
Series solution of second order ordinary differential equations with variable
coefficient
We have fully investigated solving second order linear differential equations with constant
coefficients. Now we will explore how to find solutions to second order linear differential
equations whose coefficients are not necessarily constant. Let
P(x)y'' + Q(x)y' + R(x)y = g(x)
Be a second order differential equation with P, Q, R, and g all continuous. Then x0 is a singular
point if P(x0) = 0, but Q and R do not both vanish at x0. Otherwise we say that x0 is an
ordinary point. For now, we will investigate only ordinary points.
Example
Find a solution to
y'' + xy' + y = 0 y(0) = 0 y'(0) = 1
Solution
3. Since the differential equation has non-constant coefficients, we cannot assume that a solution is
in the form y = ert
. Instead, we use the fact that the second order linear differential equation
must have a unique solution. We can express this unique solution as a power series
If we can determine the an for all n, then we know the solution. Fortunately, we can easily take
derivatives
Now we plug these into the original differential equation
We can multiply the x into the second term to get
We would like to combine like terms, but there are two problems. The first is the powers of x do
not match and the second is that the summations begin in differently. We will first deal with the
powers of x. We shift the index of the first summation by letting
u = n - 2 n = u + 2
We arrive at
Since u is a dummy variable, we can call it n instead to get
4. Next we deal with the second issue. The second summation begins at 1 while the first and third
begin at 0. We deal with this by pulling out the 0th
term. We plug in 0 into the first and third
series to get
(0 + 2)(0 + 1)a0+2x0 = 2a2
and
a0x0
= a0
We can write the series as
The initial conditions give us that
a0 = 0 and a1 = 1
Now we equate coefficients. The terms in the series begin with the first power of x, hence the
constant term gives us
2a2 + a0 = 0
Since a0 = 0, so is a2. Now the coefficient in front of xn
is zero for all n. We have
(n + 2)(n + 1)an+2 + (n + 1)an = 0
Solving for an+2 gives
-an
an+2 =
n+2
We immediately see that
an = 0
for n even. Now compute the odd an
5. -1 1 -1
a1 = 1 a3 = a5 = a7 =
3 3.
5 3.
5.
7
In general
(-1)n
2n
(n!)(-1)n
a2n+1 = =
3.
5.
7.
....
(2n+1) (2n + 1)!
The final solution is
This cannot be written in terms of elementary functions, however a computer can graph or
calculate a value with as many decimal places as needed.
Bessel equations and their series solutions
The linear second order ordinary differential equation of type
is called the Bessel equation. The number v is called the order of the Bessel equation.
The given differential equation is named after the German mathematician and astronomer
Friedrich Wilhelm Bessel who studied this equation in detail and showed (in 1824) that its
solutions are expressed through a special class of functions called cylinder functions or Bessel
functions.
Concrete representation of the general solution depends on the number v. Further we consider
separately two cases:
• The order v is non-integer;
• The order v is an integer.
Case 1. The Order v is Non-Integer
Assuming that the number v is non-integer and positive, the general solution of the Bessel
equation can be written as
6. where C1, C2 are arbitrary constants and Jv(x), J−v(x) are Bessel functions of the first kind.
The Bessel function can be represented by a series, the terms of which are expressed through the
so-called Gamma function:
The Gamma function is the generalization of the factorial function from integers to all real
numbers. It has, in particular, the following properties:
The Bessel functions of the negative order (−v) (it's assumed that v > 0) are written in similar
way:
The Bessel functions can be calculated in most mathematical software packages. For example,
the Bessel functions of the 1st kind of orders v = 0 to v = 4 are shown in Figure 1. The
corresponding functions are also available in MS Excel.
Fig.1 Fig.2
Case 2. The Order v is an Integer
If the order v of the Bessel differential equation is an integer, the Bessel functions Jv(x) and
J−v(x) can become dependent from each other. In this case the general solution is described by
another formula:
7. where Yv(x) is the Bessel function of the second kind. Sometimes this family of functions is also
called Neumann functions or Weber functions.
The Bessel function of the second kind Yv(x) can be expressed through the Bessel functions of
the first kind Jv(x) and J−v(x):
The graphs of the functions Yv(x) for several first orders v are shown above in Figure 2.
Note: Actually the general solution of the differential equation expressed through Bessel
functions of the first and second kind is valid for non-integer orders as well.
Some Differential Equations Reducible to Bessel's Equation
1. One of the well-known equations tied with the Bessel's differential equation is the modified
Bessel's equation that is obtained by replacing x to −ix. This equation has the form:
The solution of this equation can be expressed through the so-called modified Bessel functions of
the first and second kind:
where Iv(x) and Kv(x) are modified Bessel functions of the 1st and 2nd kind, respectively.
2. The Airy differential equation known in astronomy and physics has the form:
It can be also reduced to the Bessel equation. Its solution is given by the Bessel functions of the
fractional order :
3. The differential equation of type
differs from the Bessel equation only by a factor a2
before x2
and has the general solution in the
form:
8. 4. The similar differential equation
is reduced to the Bessel equation
by using the substitution
Here the parameter n2
denotes:
As a result, the general solution of the differential equation is given by
The special Bessel functions are widely used in solving problems of theoretical physics, for
example in investigating
• wave propagation;
• heat conduction;
• vibrations of membranes
in the systems with cylindrical or spherical symmetry.
Legendre equations
The Legendre differential equation is the second-order ordinary differential equation
(1)
which can be rewritten
(2)
9. The above form is a special case of the so-called "associated Legendre differential equation"
corresponding to the case . The Legendre differential equation has regular singular points at
, 1, and .
If the variable is replaced by , then the Legendre differential equation becomes
(3)
derived below for the associated ( ) case.
Since the Legendre differential equation is a second-order ordinary differential equation, it has
two linearly independent solutions. A solution which is regular at finite points is called a
Legendre function of the first kind, while a solution which is singular at is called a
Legendre function of the second kind. If is an integer, the function of the first kind reduces to a
polynomial known as the Legendre polynomial.
The Legendre differential equation can be solved using the Frobenius method by making a series
expansion with ,
(4)
(5)
(6)
Plugging in,
(7)
(8)
(9)
(10)
(11)
10. (12)
(13)
(14)
so each term must vanish and
(15)
(16)
(17)
Therefore,
(18)
(19)
(20)
(21)
(22)
so the even solution is
(23)
Similarly, the odd solution is
(24)
If is an even integer, the series reduces to a polynomial of degree with only even powers
of and the series diverges If is an odd integer, the series reduces to a polynomial of
degree with only odd powers of and the series diverges The general solution for an
integer is then given by the Legendre polynomials
11. (25)
(26)
where is chosen so as to yield the normalization and is a hypergeometric
function.
A generalization of the Legendre differential equation is known as the associated Legendre
differential equation.
Moon and Spencer (1961, p. 155) call the differential equation
(27)
Properties of Bessel function
For integer order α = n, Jn is often defined via a Laurent series for a generating function:
an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by
contour integration or other methods.) Another important relation for integer orders is the
Jacobi–Anger expansion:
and
which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series
of a tone-modulated FM signal.
12. More generally, a series
is called Neumann expansion of ƒ. The coefficients for ν = 0 have the explicit form
where Ok is Neumann's polynomial.[33]
Selected functions admit the special representation
with
due to the orthogonality relation
More generally, if ƒ has a branch-point near the origin of such a nature that
then
or
13. where is f's Laplace transform.[34]
Another way to define the Bessel functions is the Poisson representation formula and the Mehler-
Sonine formula:
where ν > −1/2 and z ∈ C.[35]
This formula is useful especially when working with Fourier
transforms.
The functions Jα, Yα, Hα
(1)
, and Hα
(2)
all satisfy the recurrence relations:
where Z denotes J, Y, H(1)
, or H(2)
. (These two identities are often combined, e.g. added or
subtracted, to yield various other relations.) In this way, for example, one can compute Bessel
functions of higher orders (or higher derivatives) given the values at lower orders (or lower
derivatives). In particular, it follows that:
Modified Bessel functions follow similar relations :
14. and
The recurrence relation reads
where Cα denotes Iα or eαπi
Kα. These recurrence relations are useful for discrete diffusion
problems.
Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions
must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it
follows that:
where α > −1, δm,n is the Kronecker delta, and uα, m is the m-th zero of Jα(x). This orthogonality
relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function
is expanded in the basis of the functions Jα(x uα, m) for fixed α and varying m.
An analogous relationship for the spherical Bessel functions follows immediately:
Another orthogonality relation is the closure equation:[36]
for α > −1/2 and where δ is the Dirac delta function. This property is used to construct an
arbitrary function from a series of Bessel functions by means of the Hankel transform. For the
spherical Bessel functions the orthogonality relation is:
15. for α > −1.
Another important property of Bessel's equations, which follows from Abel's identity, involves
the Wronskian of the solutions:
where Aα and Bα are any two solutions of Bessel's equation, and Cα is a constant independent of
x (which depends on α and on the particular Bessel functions considered). For example, if Aα =
Jα and Bα = Yα, then Cα is 2/π. This also holds for the modified Bessel functions; for example, if
Aα = Iα and Bα = Kα, then Cα is −1.
Properties of Legendre polynomials
Generating function
Let F(x t) be a function of the two variables x and t that can be expressed as a Taylor’s series in
t, ∑ncn(x)tn. The function F is then called a generating function of the functions cn.
Example 11.1:
Show that F(x t)=11−xt is a generating function of the polynomials xn.
Solution:
Look at
11−xt=∑∞n=0xntn( xt 1) (11.16)
Example 11.2:
Show that F(x t)=exp 2ttx−t is the generating function for the Bessel functions,
F(x t)=exp(2ttx−t)=∑∞n=0Jn(x)tn (11.17)
Example 11.3:
16. (The case of most interest here)
F(x t)=1 1−2xt+t2=∑∞n=0Pn(x)tn (11.18)
Rodrigues’ Formula
Pn(x)=12nn!dndxn(x2−1)n (11.19)
A table of properties
1. Pn(x) is even or odd if n is even or odd.
2. Pn(1)=1.
3. Pn(−1)=(−1)n.
4. (2n+1)Pn(x)=P n+1(x)−P n−1(x).
5. (2n+1)xPn(x)=(n+1)Pn+1(x)+nPn−1(x).
6. ∫x−1Pn(x )dx =12n+1 Pn+1(x)−Pn−1(x) .
Let us prove some of these relations, first Rodrigues’ formula. We start from the simple formula
(x2−1)ddx(x2−1)n−2nx(x2−1)n=0 (11.20)
which is easily proven by explicit differentiation. This is then differentiated n+1 times,
dn+1dxn+1 (x2−1)ddx(x2−1)n−2nx(x2−1)n
===n(n+1)dndxn(x2−1)n+2(n+1)xdn+1dxn+1(x2−1)n+(x2−1)dn+2dxn+2(x2−1)n−2n(n+1)dndx
n(x2−1)n−2nxdn+1dxn+1(x2−1)n−n(n+1)dndxn(x2−1)n+2xdn+1dxn+1(x2−1)n+(x2−1)dn+2dxn
+2(x2−1)n− ddx(1−x2)ddx dndxn(x2−1)n +n(n+1) dndxn(x2−1)n =0
We have thus proven that dndxn(x2−1)n satisfies Legendre’s equation. The normalisation
follows from the evaluation of the highest coefficient,
dndxnx2n=n!2n!xn (11.22)
and thus we need to multiply the derivative with 12nn! to get the properly normalised Pn.
Let’s use the generating function to prove some of the other properties: 2.:
17. F(1 t)=11−t=∑ntn (11.23)
has all coefficients one, so Pn(1)=1. Similarly for 3.:
F(−1 t)=11+t=∑n(−1)ntn (11.24)
Property 5. can be found by differentiating the generating function with respect to t:
ddt1
1−2tx+t2x−t(1−2tx+t2)3∕2x−t1−2xt+t2∑∞n=0tnPn(x)∑∞n=0tnxPn(x)−∑∞n=0tn+1Pn(x)∑∞n=0t
n(2n+1)xPn(x)=====ddt∑∞n=0tnPn(x)∑n=0ntn−1Pn(x)∑n=0ntn−1Pn(x)∑∞n=0ntn−1Pn(x)−2
∑∞n=0ntnxPn(x)+∑∞n=0ntn+1Pn(x)∑∞n=0(n+1)tnPn+1(x)+∑∞n=0ntnPn−1(x)
Equating terms with identical powers of t we find
(2n+1)xPn(x)=(n+1)Pn+1(x)+nPn−1(x) (11.26)