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Answered: - Differential equation practice exam, need the solutions.


Differential equation practice exam, need the solutions.


Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

Show all of your work.

 


 

1. For the matrix equation below,

 

()? = []() + ()

 

a. What will be true for the equation to be homogeneous?

 


 

b. What will be true for the equation to be non-homogeneous?

 


 

2.

 


 

Write the system of equations as a matrix equation and initial condition vector.

 

?

 

1 ? 2 + 61 ? 3 =

 

?

 

2 = 31 ? 93 ? 42

 

?

 

3 + 21 ? 2 = 0

 


 

1 (0) = 0 , 2 (0) = 2 , 3 (0) = 0

 


 

1

 


 

Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

3. Consider the following second-order equation:

 

?? + ? + 5 = 2

 

a. Rewrite the above equation as a system of first order equations. Use 1 and 2 as the

 

dependent variables and t as the independent variable.

 


 

b. Write the system of equations from part (a) as a single matrix equation.

 


 

2

 


 

Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

4. Consider the system of first-order equations.

 

?

 

1 ? 21 ? 22 = 0

 

?

 

2 ? 1 ? 32 = 0

 


 

a. Find the eigenvalues and eigenvectors for the system. (Select values of ?s? such that the

 

eigenvectors have the lowest integer-valued elements.)

 


 

b. Write a general solution vector.

 


 

3

 


 

Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

5. Consider the following system of equations and initial conditions:

 

?

 

1 = ?61 + 52

 

?

 

2 = ?51 + 42

 


 

1 (0) = 0 , 2 (0) = 1

 

a. Find the solution vector to the initial value problem.

 


 

4

 


 

Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

b. Write the solution to part (a) as a set of scalar equations.

 


 

c. Use the Wronskian to show that the two vector solutions in part (a) make up a fundamental

 

solution set.

 


 

5

 


 

Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

6. Consider the following system of equations and initial conditions:

 

?

 

1 = 41 + 52

 

?

 

2 = ?21 + 62

 


 

1 (0) = 1 , 2 (0) = 0

 

Find the solution vector to the initial value problem.

 


 

6

 


 

Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

7. Write the governing system of first order equations for the following problem.

 

Two large tanks, each holding 24 liters of a brine solution, are interconnected by pipes as shown in

 

the figure. Brine (1 kg/L) flows into tank A at a rate of 6 L/min, and fluid is drained out of tank B at

 

the same rate; also 8 L/min of fluid are pumped from tank A to tank B, and 2 L/min from tank B to

 

tank A. The liquids inside each tank are kept well stirred so that each mixture is homogeneous.

 

Write a first-order matrix equation that describes this system. (You do not need to solve the

 

equation.)

 


 

7

 


 

Math 2500: Differential Equations

 

Exam #3 ? Spring 2016 Practice

 

Instructor: Mel Henriksen

 


 

Name:

 


 

8. Derive the requested equations below.

 

a. For a second-order, linear, constant-coefficient equation such as

 

?? + ? + = 0

 

assume that the following function is a solution:

 

=

 

Derive the auxiliary equation.

 


 

b. For a system of first-order linear, constant-coefficient equations expressed as a single matrix

 

equation such as:

 

? = []

 

assume that the following vector function is a solution:

 

=

 

Derive the two necessary criteria (equations) that can be used to calculate the eigenvalues

 

and eigenvectors.

 


 

8

 


 

 


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