Math 285  Practice Final Exam    -   Spring 2015

 

1.  Solve the following initial value problem for y(x).

 

        ( y / (2x ˗ 1)  dy/dx  ˗  y2  =  1        with  y(0)  =  2

 

Answer:    y2  =  5  exp( 2x2 ˗ 2x)    ˗  1

 

 

2.  Solve the initial value problem for P(t).  Sketch several solution curves.  Indicate

     the equilibrium solutions and their stability.  Draw the solution curve for the initial

     condition.

 

               dP/dt  =  P2   +  3P  ˗ 4         with  P(0)  =  2

 

Answer:    (P ˗ 1) / (P + 4)  =  (1/6) e 5t 

 

 

3.  Find the general solution y(x) for the following D.E.

 

                   y '''  +  y ''  ˗ 2 y '  =  2 x2    

                                   

 Answer:  y(x)  =   A  + B ex  +  C e˗ 2x  ˗ (1/3) x3  ˗ (1/2) x2  ˗ (3/2) x

 

 

4.  Calculate all non-negative eigenvalues and corresponding eigenfunctions for the

     following boundary value problem.

 

              y ''  +  λ y  =  0      y(0)  =  y '(3)  =  0

 

Answers:             λn  =  ( nπ/6)2     yn(x)  =  sin (nπx/6)

 

 

5.  Find the general solution for the following equation.

                                                   ∞

                    x ''  +  9 x   =  2  +  ∑ (1/n3)  cos (nt)

                                                  n=1

                                                                               ∞

Answer:  y(x)  =  A cos 3t  +  B sin 3t  +  2/9  +  ∑ 1/(n3(9 ˗ n2) cos nt  +  (1/162) t sin 3t

                                                                            n=1  and n≠3

 

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