Differential Equations Exam #2
Fall 2003
Name_________________________________
Show all your work neatly and in numerical order on notebook paper. You must omit one problem by clearly writing
“OMIT” by the problem on your notebook paper. If you do not omit a problem, I will omit the last one for you.
DO NOT WRITE ON THE BACKS OF YOUR PAGES.
1. a. Define “fundamental set of solutions”. Be very precise.
b. We know that a fundamental set of solutions exists for what type of differential equations? What does this type of
differential equation look like?


y   4 y  x 2  3 sin 2 x , yc  c1 cos 2 x  c2 sin 2 x . Find the assumed form of the
particular solution y p when using undetermined coefficients. (Do not find y p .)
2. For the differential equation
3. Suppose
m1  1 is a root of multiplicity 3 and m2  1  2i, m3  1  2i are each roots of multiplicity 1 of an
auxiliary equation. Write down the general solution of the corresponding homogeneous linear DE if it is a CauchyEuler equation.
4. Solve: y   8 y   16 y  0 .
5. Solve:
x 2 y   3xy   2 y  0
6. When solving
y   y   12 y  e 4 x using undetermined coefficients you will get y c  c1e 4 x  c 2 e 3 x . Find y p and
write the general solution of the differential equation.
7. Solve
8. Solve:
y   y  cos 2 x given that yc  c1 cos x  c2 sin x .
dx dy

 2x  2 y  1
dt dt
dx
dy
2
 y3
dt
dt
9. Solve y    y   1  0 by using the substitution u  y  .
2
x 2 y   xy   y  0 is y  c1 x  c2 x ln x, (0, ) . Find the particular solution that satisfies
the initial conditions y (1)  3, y (1)  1 .
10. The general solution of
11. a. Without the aid of the Wronskian, determine whether the given set of functions is linearly independent or linearly
dependent on the indicated interval.
f1 ( x)  ln x,
f 2 ( x)  ln x 2 , (0, )
b. Now use the Wronskian to determine whether the functions given in part (a) are linearly independent or linearly
dependent. What significance does this have for us as we solve differential equations?