Fourth Order Homogenous Differential Equation

[shards5]

Homework Statement

y⁴ - 6y³ + 9y²
y(0) = 19; y'(0) = 16; y"(0) = 9; y"'(0) = 0

Homework Equations

N/A

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*e^3x + k*x*e^3x + L*k*x²*e^3x
I am assuming since one of the roots is zero then the solution will not have to have add i*e^0t. 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?

 

[Mark44]

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, e^3x, xe^3x}. Your general solution will be all linear combinations of these functions, or
y(x) = c₁*1 + c₂*x + c₃*e^3x + c₄*xe^3x. Use your initial conditions to solve for the constants c_i.

 

[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.