This document contains a 4 question exam for an applied ordinary differential equations course. The exam tests skills in: [1] finding general solutions to differential equations in implicit form; [2] solving Bernoulli equations by substitution; [3] solving exact initial value problems in implicit form; and [4] modeling and solving a word problem about salt levels in a mixing tank over time. The exam is 60 minutes and uses an admissible calculator, with no cell phones allowed.
Differential Equations Lecture: Non-Homogeneous Linear Differential Equationsbullardcr
A lecture I presented in Differential Equations, Spring 2006. This was supplemented with a hands-on solution to a random problem with variables designated by students in the class.
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:
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.
I am Eugeny G. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Columbia University. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
Differential Equations Lecture: Non-Homogeneous Linear Differential Equationsbullardcr
A lecture I presented in Differential Equations, Spring 2006. This was supplemented with a hands-on solution to a random problem with variables designated by students in the class.
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:
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.
I am Eugeny G. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Columbia University. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
1. Concordia University February 13, 2009
Applied Ordinary Differential Equations
ENGR 213 - Section J
Prof. Alina Stancu
Exam I (A)
Directions: You have 60 minutes to solve the following 4 problems. You may use an admis-
sible calculator. No cell phones are allowed during the exam.
(1) (10 points) Find a general solution of the differential equation
dy
= 4(y 2 + 1).
dx
You may leave the solution in implicit form.
(2) (10 points) Solve the following Bernoulli equation by the appropriate substitution
dy
+ y = xy 4 .
dx
You may leave the solution in implicit form.
(3) (10 points) Solve the exact initial value problem
x 3y 2 − x2
dx + + 2y dy = 0, y(0) = 2,
2y 4 y5
leaving the solution in implicit form.
(4) (10 points) A large tank is filled to capacity with 100 gallons of pure water. Brine
containing 3 pounds of salt per gallon is pumped into the tank at a rate of 4 gal/min.
The well-mixed solution is pumped out of the tank at the rate of 5 gal/min.
Find the number of pounds of salt in the tank after 30 minutes.
1