practice2 234 key - The Citadel · Microsoft Word - practice2_234_key.docx Author: twittman Created Date: 3/10/2015 2:12:53 PM ...

Math 234 Practice Exam 2 Key

1.) [10 points] Find the indicated Laplace Transform.

a.) ℒ�4�� − 6� − 14

b.) ℒ�3�� cos�2�� − 42����

c.) ℒ�sin�3�� ∗ cosh�5��

d.) ℒ���sin�3��

e.) ℒ�cos�4�� ��� − 1�

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2.) [10 points] Find the indicated Inverse Laplace Transform. Do not use convolutions.

a.) ℒ�� � ! −�

!"�#+ 7���!&

b.) ℒ�� � �!���!� ��!'��&

c.) ℒ�� ��()*+!"'# &

d.) ℒ�� � !!"',!'�&

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3.) [10 points] Charizard performed the calculations shown below.

Explain what is wrong with Charizard's calculation and then show the

correct way to do the problem.

a.) Compute ℒ��� sin�4��.

ℒ��� sin�4�� - ℒ��� ℒ�sin�4�� - . 1/ − 30 .

4/� + 160 - 4

�/ − 3��/� + 16�

Laplace Transforms do not break up under products: ℒ�12 3 ℒ�1ℒ�2. This property only works for convolutions: ℒ�1 ∗ 2 - ℒ�1ℒ�2.

Use Rule 19 with a=3, b=4:

b.) Compute ℒ �!()4+!"���&. ℒ�� ���,! !

!"���& - ��� − 6�ℒ�� � !!"���& - ��� − 6� cosh�5��

Rule 27 says we have to shift the function 1��� to 1�� − 6�.

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4.) [20 points] Solve the initial value problem:

566 − 256 − 35 - 4��� − 2�, 5�0� - 2,56�0� - 0