Linear dierential equation

1. The functions f1(x) = 5x; f2(x) = x
   2 + 3x, and f3(x) = 2x
   2 + x do not form a linearly
   independent set of functions. Verify this by writing one of the given functions as a linear
   combination of the other two functions. (2 pts.)
2. Show that the functions f1(x) = e
   4x
   and f2(x) = e
   2x
   form a linearly independent set of
   functions. (2 pts.)
3. Suppose that the auxiliary equation of a fth-order homogeneous linear dierential equation
   with constant coecients has roots m1 = 1; m2 = 1; m3 = 4; m4 = 2 + i, and
   m5 = 2 i. What is the solution to the dierential equation? (2 pts.)
4. Solve y
   000 y
   00 9y
   0 + 9y = 0. (4 pts.)
5. Use the method of undetermined coecients to solve y
   00 6y
   0 + 8y = 2x+ 1 + 4e
   2x
   . (5 pts.)
6. Use variation of parameters to solve y
   00 4y
   0 + 3y = e
   5x
   . (5 pts.

Sample Solution