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# MATH 204 PRACTICE MIDTERM II

Please work out each of the given problems on your own paper.  Credit will be based on the steps that you show towards the final answer.  Show your work.

Problem 1

Solve the following differential equations.

A.  y'' - 2y' + 2y  =  0,  y(0)  =  0,  y'(0)  =  5

Solution

B.  y''' - 3y'' + 3y' - y  =  0

Solution

C.  y(iv) - y  =  0

Problem 2
Consider the differential equation

y''' - y'  =  e2t + e3t

A.     Solve this differential equation using the method of UC functions.

Solution

B.     Solve this differential equation using the method of variation of parameters.

Solution

Problem 3

A 2 kg mass stretches a spring 0.0784 meters.  The mass is attached to a viscous damper that exerts a force of 4N when the velocity is 0.1 m/sec.  The mass is then pulled down 0.5m and released.

A.     Determine the equation of motion for this system.

B.     Describe (qualitatively) the difference between replacing the viscous damper with an external force of F  =  3sin(25t) and replacing the damper with an external force of F  =  2cos(3t).

Problem 4

Please answer the following true or false.  If false, explain why or provide a counter-example.  If true, explain why.

A.     The differential equation

(t - 1)y'' + cos t y' + (t - 1)y  =  et,    y(0)  =  3,     y'(0)  =  4

has a unique solution defined for all real numbers.

B.     Let y1  =  t + 1y2  =  4t,  be solutions of the differential equation

y''' + p(t)y'' + q(t)y' + r(t)y  =  s(t)

with p, q, r, and s all continuous.  Then

y3  =  sin t

cannot also be a solution of this differential equation.

Two Important Formulas:

1.    mu'' + gu' + ku  =  F

2.   LQ'' + RQ' + 1/C Q  =  E'(t)