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MAT 204 Practice Exam 2

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.  

Printable Key

Problem 1 

Solve the following differential equations.

A.  y''' - 3y'' + 3y' - y  =  0

B.  y^(iv) - y  =  0  

Problem 2 
Consider the differential equation

        y''' - y'  =  e^2t + e^3t

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

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

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 
Solve the given differential equation by means of a power series about x = 0.  Find a recurrence relation and write the final solution in the form

        y'' + xy' + 2y  =  0,        y(0)  =  0,     y'(0)  =  1

Problem 5 

Determine the general solution of the differential equation that is valid in any interval not including the singular point. 

        x²y'' - xy' + y  =  0      

Problem 6 

Solve the following differential equation

            y(0)  =  y'(0)  =  0  

Problem 7  Please answer the following true or false.  If true, explain why.  If false, explain why or provide a counter-example.

A.     The differential equation

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

has a unique solution defined for all real numbers.  

B.     Let y₁  =  t + 1,  y₂  =  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.

C.    If  f(x)  is a function that is not continuous at x  =  2 , then the Laplace transform of  f(x)  is also not continuous at x =  2 .  

D.  Let      be a solution of the differential equation

            x                     1  
              y''  +                y'  + (sin x)y  =  0
    x + 4              x - 2                

 then  x  =  -1  is in the interval of convergence of y(x).