Differential Equations  Practice Final Exam    -   Math 285   Spring 2011

 

 

 

 

 

 

 

 

 

 

1.

 

 This problem has two parts, each concerning a particular differential equation. 

 

a.        Consider a population of rabbits, initially numbering 100, whose growth is

 governed by the equation

 

               dP/dt  =  P2 / 250  -  4P / 5

 

What is the long-term behavior of this population?    i.e.  lim   P(t)?

                                                                                            tà ∞

b.      A copper wire of length 100 cm is heated to a uniform temperature of 100oC,

and then at time  t = 0 one of its ends is placed on ice (0oC), while the other end

Is kept at 100oC throughout.  The heat flow in the wire satisfies the heat equation

ut  =  kuxx.  What is the steady-state temperature distribution (i.e. temperature

distribution as  t à ∞ ?

 

Hint:  Both questions can be answered without explicitly solving the relevant equation.

 

Answer:  a.  0           b.  100 x

 

 

 

2.

Solve the initial value problem below.       Hint:  Find integrating factor

 

                   (x4 + 1) dy/dx  + 2 x3 y =  x3               y(0)  =  5

                                   

                    Answer:  y(x)  =  ½  +  9/ [ 2√(x4 + 1) ]

 

 

 

 

 

 

3.

 

 

Find the general solution to the two equations below.  (For both equations, your answer

for y(x) should be in explicit rather than implicit form.)

 

a.                 dy/dx  = xy2  + 3xy  + 2x

 

b.                 xy dy/dx  =  x2  +  3y2

 

Answer:   a.  y(x)  =  1 / [ 1 – C exp( x2/2) ] – 2

 

                 b.  y2  =  x2 [ Cx4  –  1/2 ]

 

 

 

 

4.

 

Use variation of parameters to solve the equation below.

                                        

                    y’’ +  4y  =  sin 2x

 

Answer:  y(x)  =  - (x/4) cos 2x  +  C1 cos 2x  +  (C2 + 1/16) sin 2x

 

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