math 234, spring 2014 final exam keymacs.citadel.edu/wittman/234/exams/final_234_14s_key.pdf ·...
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![Page 1: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/1.jpg)
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
![Page 2: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/2.jpg)
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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
![Page 3: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/3.jpg)
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#2 continued...
−
−
=
243
043
121
A ,
−=
213
602B
d.) BBT
e.) || A
f.) || B
![Page 4: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/4.jpg)
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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�)
![Page 5: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/5.jpg)
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4.) [10 points] Find the general solution of the second-order differential equation:
��� − 4�� + 8� = 40�� − 40�
![Page 6: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/6.jpg)
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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)�
![Page 7: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/7.jpg)
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6.) [10 points] Find the indicated Inverse Laplace Transform.
a.) ℒ�" # $%�&$'""(
b.) ℒ�" # $')+
$%')(
c.) ℒ�" # *+,-
$%�.(
![Page 8: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/8.jpg)
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7.) [10 points] Solve the initial value problem
��� − 4�� + 3� = 2/(� − 10), �(0) = 2,��(0) = 0
![Page 9: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/9.jpg)
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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.
![Page 10: Math 234, Spring 2014 Final Exam Keymacs.citadel.edu/wittman/234/Exams/final_234_14s_key.pdf · Math 234, Spring 2014 Final Exam Key 1.) [10 points] Find the solution of the system,](https://reader034.vdocuments.net/reader034/viewer/2022050518/5fa2590c6c9f571ef27ecaa2/html5/thumbnails/10.jpg)
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9.) [10 points] Find the general solution of the system of differential equations
�2� = 35 −51 3 4 �2