# 3rd Order Linear Homogenous DE from Solutions

1. Apr 12, 2012

### SArnab

1. The problem statement, all variables and given/known data

Find a third order, linear, homogeneous DE which has the following solutions:

e$\pi$t, te$\pi$t and e-t

2. Relevant equations

Standard form of a third-order linear homogenous ODE:

Ay''' + By'' + Cy' + Dy = 0

3. The attempt at a solution

I tried deriving the characteristic equation given the r values of the solutions.

For e$\pi$t:
r = $\pi$, therefore one part of the equation is (r - $\pi$)

For e-t:
r = -1, therefore another part of the equation is (r+1)

But I can't figure out what to do for the second one.

Last edited: Apr 12, 2012
2. Apr 12, 2012

### LCKurtz

Remember that a repeated root $r$ of the characteristic equation gives rise to the solution pair $\{e^{rt},te^{rt}\}$.

3. Apr 12, 2012

### SArnab

Ah, so it just has a repeated root.

So the characteristic equation would be:

(r-$\pi$)2(r+1)

Factor that out and we get:
(r2 - 2r$\pi$ + $\pi$2)(r+1)

r3 - 2r2$\pi$ + r$\pi$2 + r2 - 2r$\pi$ + $\pi$2

r3 - 2r2$\pi$ + r2 + r$\pi$2 - 2r$\pi$ + $\pi$2

r3 + r2(-2$\pi$ + 1) + r($\pi$2 - 2$\pi$) + $\pi$2

Therefore the DE would be:

y''' + (1-2$\pi$)y'' + ($\pi$2 - 2$\pi$)y' + $\pi$2y = 0

Thanks for the help!