Linear di erential 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 di erential 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 di erential 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

ACED ESSAYS