Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system, if it exists. Show all work.

12 321 −=++ xxx

332 31 =− xx

2223 321 =−+ xxx

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2.) [20 points] Given the following matrices.

−

−

=

243

043

121

A ,

−=

213

602B

Compute following quantities, if they exist. If the quantity is undefined, explain why.

a.) BA +

b.) AB

c.) BA

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#2 continued...

−

−

=

243

043

121

A ,

−=

213

602B

d.) BBT

e.) || A

f.) || B

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3.) [10 points] Find an explicit solution for the following first-order differential equations.

a.) ���� + 3�� = 5� − 2

b.) �� = x�sec(3�)

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4.) [10 points] Find the general solution of the second-order differential equation:

��� − 4�� + 8� = 40�� − 40�

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5.) [10 points] Find the indicated Laplace Transform.

a.) ℒ�2� + ��� cos(4�)�

b.) ℒ��� sin(3�)�

c.) ℒ�cos(6�)!(� − 4)�

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6.) [10 points] Find the indicated Inverse Laplace Transform.

a.) ℒ�" # $%�&$'""(

b.) ℒ�" # $')+

$%')(

c.) ℒ�" # *+,-

$%�.(

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7.) [10 points] Solve the initial value problem

��� − 4�� + 3� = 2/(� − 10), �(0) = 2,��(0) = 0

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8.) [10 points] A 2x2 matrix A has the eigenvalues and eigenvectors below.

3 1 −=λ ,

=

1

51vv

, 42 =λ ,

−=

1

12vv

a.) Write the general solution of the system of differential equations xAxvv

=′ .

c.) Classify the origin as an attractor, repeller, or saddle point and sketch the phase portrait.

c.) Find the specific solution �2 that satisfies the initial condition �2(0) = 3−23 4.

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9.) [10 points] Find the general solution of the system of differential equations

�2� = 35 −51 3 4 �2