## 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:

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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