# Thread: Changing equation into ODE... easy?

1. ## Changing equation into ODE... easy?

Find an ODE of the form $\displaystyle y''+ay'+by=0$ where $\displaystyle a^2>4b$ for which the given function is a solution, or else explain why no such ODE exists.

The first ones I have been able to figure out by reverse-engineering the solving process...

e.g. $\displaystyle e^t+exp(2t) \longrightarrow y''-3y'+2y=0$

However, I am having trouble with these two:
• $\displaystyle t^2 e^t$
• $\displaystyle exp(x^2)$

Any insight would be greatly appreciated!

2. Originally Posted by horan
Find an ODE of the form $\displaystyle y''+ay'+by=0$ where $\displaystyle a^2>4b$ for which the given function is a solution, or else explain why no such ODE exists.

The first ones I have been able to figure out by reverse-engineering the solving process...

e.g. $\displaystyle e^t+exp(2t) \longrightarrow y''-3y'+2y=0$

However, I am having trouble with these two:
• $\displaystyle t^2 e^t$
• $\displaystyle exp(x^2)$

Any insight would be greatly appreciated!
say for $\displaystyle t^2e^t$

let $\displaystyle y = t^2 e^t$

find $\displaystyle y'$ and $\displaystyle y''$ then plug them into the form of the equation you were given. then solve for $\displaystyle a,b$ such that $\displaystyle a^2 > 4b$

3. Thanks for the reply!. Unfortunately, this yields a really messy slew of algebra :-(

I get:
$\displaystyle 4 t e^t+e^t(2+t^2)+a t e^t (t+2)+b t^2 e^t = 0$

... but I cannot figure out how to prove that $\displaystyle a^2>4b$.