Mathematics 38 Differential Equations

Final Exam May 9, 2011 8:30 - 10:30 AM

No calculators, books or notes are allowed on the exam. All electronic devices must be turned off and put away.

You must show all your work in the blue book in order to receive full credit. Please box your answers and cross out

any work you do not want graded. Make sure to sign your blue book. With your signature you are pledging that you

have neither given nor received assistance on the exam. Good luck!

1. (30 points, 5 each) No partial credit.

a. Check for independence of:

1) the collection of functions x, x ln x, x ln x2;

2) the collection of vectors

1

2

3

4

,

8

7

6

5

,

13

8

3

−2

,

12

10

4

−1

;

b. Use the deﬁnition of the Laplace transform to compute L[te−t];

c . Find (D2+ 2D+ 1)[e2tsin t];

d. Evaluate et∗t;

e . Find all solutions of the equation

x

′+x=x2;

f . Solve the initial value problem

x

′+x=x2, x(0) = 1

4.

2. (6 points) Find the general solution of the non-homogeneous equation

x

′+xtan t= sin 2t.

3. (10 points) Consider the following initial value problem:

x

′=p|x|, x(0) = 0.

a. Is the existence and uniqueness theorem applicable?

b. If it is not applicable, does the IVP has a solution?

c . If a solution exists, is it unique? Explain.

Examination continues on other side

## Document Summary

No calculators, books or notes are allowed on the exam. All electronic devices must be turned off and put away. You must show all your work in the blue book in order to receive full credit. Please box your answers and cross out any work you do not want graded. With your signature you are pledging that you have neither given nor received assistance on the exam. Good luck: (30 points, 5 each) no partial credit. a . Check for independence of: the collection of functions x, x ln x, x ln x2, the collection of vectors. 1: use the de nition of the laplace transform to compute l [te t]; c . Find (d2 + 2d + 1)[e2t sin t]: evaluate et t; e . Find all solutions of the equation f . 4: (6 points) find the general solution of the non(cid:173)homogeneous equation x.