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MATH 239 DIFFERENTIAL EQUATIONS
FALL 2008 TEST 1
NAME_______________________________
Show all your work and please box your final answer.
1. Which of the following functions is a solution to y' '2 y 3  0 ?
(Show work to support your answers.)
y1  e 2t
y2  sin( t )
y3 
1
t
For the solution you just found, is a constant multiple of that solution also a solution?
2. Which of the following functions is a solution to y ' '2 y '8 y  0 ?
(Show work to support your answers.)
y1  e 2t
y2  sin( t )
y3 
1
t
For the solution you just found, is a constant multiple of that solution also a solution?
There's an important difference between the differential equations in 1 and 2. What is that
difference?
3. Make up an interesting exact differential equation and prove that it is exact.
4. Find and classify all equilibrium solutions of y   y 2 ( y  4) 2 ( y  6) and make a sketch of possible
solutions (direction field). Based on your sketch, determine the behavior of y as t   and if this
behavior depends on the initial value of y at t  0 , describe this dependency.
5. Consider the first order linear differential equation y   y  t  2 .
a) Create a slope field for this diff eq and based on the slope field, explain the end behavior of
solutions and/or how the end behavior depends on the choice of initial conditions.
b) Now actually solve the diff eq and confirm your conclusion about end behavior by calculating
lim t  y .
6. Solve the following differential equations, incorporating any given initial conditions. There's one
here of each type we know how to solve, trust me.
a)
dy
 cos( x) y  8 cos( x),
dx
y (0)  10
y

b)   6 x dx  (ln( x)  2)dy  0
x

c) 2 sin( y ) cos( x)dx  cos( y ) sin( x)dy  0
7. Take a look at this differential equation: ( y   y )t  y ' for t  0 .
Its second order, so as it is we can't solve it using the methods we've seen so far. But the
substitution v  y  and v'  y  will make it first order in v .
a) Do this substitution and solve the first order differential equation to get a general solution for v .
b) Now refer back to your original substitution to find a general solution for y .
: : Bonus : :
If a function is a quadratic, that implies it’s a polynomial. If something's a square, that implies
that it's a rectangle. Do any of the types we focused on in Chapter 2 (linear, exact, separable)
imply any other type? Provide some justification for your answer.