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**preview**shows half of the first page. to view the full**2 pages of the document.**Mathematics 51 Differential Equations

Examination III May 6, 2013, 8:30â€“10:30am

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. (10 points)

a. Show that x(t) = t+ 4 is a solution of the differential equation sin(t)D3x+4Dxâˆ’x=âˆ’t.

b. Write sin(t)D3x+ 4Dx âˆ’x=âˆ’tas a system of differential equations.

c . Give

one

solution to the system

in vector form

.

2. (10 points) Solve the initial-value problem dx

dt =x2,x(0) = âˆ’3.

3. (10 points) Find all solutions (

in vector form

) to the equation D~x =ï£«

ï£¬

ï£

5 3 0 0

âˆ’3âˆ’1 0 0

0 0 1 0

0 0 0 7

ï£¶

ï£·

ï£¸~x.

4. (10 points) Find all solutions to the equation D~x =î€’3 0

0 5 î€“~x +î€’1

etî€“

5. (10 points)

Use the Laplace transform

to solve the initial value problem D2x+Dx âˆ’2x= 2

with x(0) = âˆ’1and xâ€²(0) = 0. No credit by any other method.

6. (10 points) Find all solutions to the differential equation ((Dâˆ’1)2+ 1)(D+ 3)Dx = 12.

Examination continues on next page

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