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M251ex2(fa02)

# M251ex2(fa02) - the damping coeFcient γ will make the...

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MATH 251 2nd exam Nov. 12, 2002 Name: Student Number: Instructor: Section: There are 9 partial credit questions. In order to obtain full credit for these prob- lems, all work must be shown. Credit will not be given for an answer not sup- ported by work. THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINA- TION. At the end of the examination, the booklet will be collected.

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MATH 251 -2nd exam- 1. Transform the given initial value problem into an initial value problem for two Frst order equations. x 2 y 00 + xy 0 + ( x 2 - λ 2 ) y = 0 , y (0) = y 0 , y 0 (0) = y 0 0 2. What is the form of the particular solution to the equation y 00 + 4 y = e 2 t + sin(2 t ) - 3 cos( t ) + t 2 - 4 DO NOT SOLVE FOR THE CONSTANTS! Page 2 of 8
MATH 251 -2nd exam- 3. (a) Sketch the function. f ( t ) = t - u 1 ( t )( t - 1) + u 3 ( t )(2 - t ) (b) Write the given function in terms of unit step functions. g ( t ) = t 2 0 t < 2 6 - t 2 t < 6 0 6 t Page 3 of 8

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MATH 251 -2nd exam- 4. A mass of 3 kg is attached to a spring with spring constant k = 12 N m . What value of

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Unformatted text preview: the damping coeFcient γ will make the system critically damped? 5. ±ind the Laplace transform. L { ( e πt sin(3 t ) 000 } HINT: µ ( e πt sin(3 t ) = πe πt sin(3 t ) + 3 e πt cos(3 t ) ( e πt sin(3 t ) 00 = ( π 2 + 9) e πt sin(3 t ) + 6 πe πt cos(3 t ) ¶ Page 4 of 8 MATH 251 -2nd exam-6. Find the solution to the following initial value problem. y 00 + 2 y-8 y = e 3 t u 3 ( t ) , y (0) = 0 , y (0) = 1 Page 5 of 8 MATH 251 -2nd exam-7. Solve the initial value problem. y 00 + 5 y + 4 y = 2 δ ( t-3) , y (0) = 1 , y (0) = 0 Page 6 of 8 MATH 251 -2nd exam-8. Find the general solution. ~x = µ 1 4 3 ¶ ~x, ~x = µ 7 ¶ . Page 7 of 8 MATH 251 -2nd exam-9. Find the general solution to the following systems of equations. (a) ~x = µ 1-3 3 1 ¶ ~x (b) ~x = µ 3 2-2-1 ¶ ~x Page 8 of 8...
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M251ex2(fa02) - the damping coeFcient γ will make the...

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