Differential Equations                                        Quiz #5                                                      April 1, 2005

Name____________________                   R.  Hammack                                                 Score ______

(1)    Verify that the set of functions e^(-3x), e^(4x) forms a fundamental set of solutions to y''-y'-12y=0.

First, let's confirm that y_1 = e^(-3x) is a solution to the D.E.
y_1' = -3e^(-3x) and y_1'' = 9e^(-3x). Plugging this in,
y''-y'-12y = 9e^(-3x)-(-3e^(-3x))-12e^(-3x) = 0, so it is a solution.

Next, let's confirm that y_1=e^(4x) is a solution to the D.E.
y_1' = 4e^(4x) and y_1'' = 16e^(4x). Plugging this in,
y''-y'-12y = 16e^(4x)-4e^(4x)-12e^(4x) = 0, so it is a solution.

Also, these functions are linearly independent, for

| e^(-3x) e^(4x) |
-3e^(-3x) 4e^(4x)

Thus, as two linearly independent solutions to a second order homogeneous D.E., these functions form a fundamental set of solutions.

(2)   Given that  y_1=x^4 is a solution to x^2y''-7x y'+16y=0, find another solution y_2 for which the set {y_1,y_2} is a fundamental set of solutions.

First, we divide through by x^2 to put this equation into a standard form.
y''-7/x y'+16/x^2y = 0

y_2 = y_1(x) ∫e^(-∫P(x) dx)/y_1^2(x) dx = x^4∫e^(-∫ -7/x dx)/(x^4)^2 ... dx = x^4∫x^7/x^8dx = x^4∫1/xdx  = x^4ln(x)    on (0, ∞)