# Fourth Order Homogenous Differential Equation

1. Nov 23, 2009

### shards5

1. The problem statement, all variables and given/known data
y4 - 6y3 + 9y2
y(0) = 19; y'(0) = 16; y"(0) = 9; y"'(0) = 0

2. Relevant equations
N/A

3. The attempt at a solution
Factored out the equation and obtained the following roots.
r2(r2-3) = 0 which gives r = 0 and r =3.
Using those roots, I make the following general solution.
y(x) = j*e3x + k*x*e3x + L*k*x2*e3x
I am assuming since one of the roots is zero then the solution will not have to have add i*e0t. I am also assuming since this is a fourth order equation that I will need to solve for n=4 variables and that this is the form in which I should tackle it. Am I mistaken in my assumptions?

2. Nov 23, 2009

### Staff: Mentor

Is your differential equation? y(4) - 6y(3) + 9y(2) = 0?

Your characteristic equation has four roots, with 0 and 3 repeated. Your basic set of solutions is {1, x, e3x, xe3x}. Your general solution will be all linear combinations of these functions, or
y(x) = c1*1 + c2*x + c3*e3x + c4*xe3x. Use your initial conditions to solve for the constants ci.

3. Nov 23, 2009

### shards5

Yea it was equal to zero and thanks for the general equation. It solved a lot of the confusion I had on what to do with the r = 0.