# [PDE] Transforming Hard Equations into Easier Ones

1. Feb 24, 2013

### Arkuski

I have a PDE and I have to transform it into an easier one using a substitution:

$u_t=u_{xx}-\beta u$

I am supposed to use the following substitution:

$u(x,t)=e^{-\beta t}w(x,t)$

I am supposed to get something that looks like this:

$w_t=w_{xx}$

Can someone show me the steps?

2. Feb 24, 2013

### Ray Vickson

Here are the steps. Substitute the form $u(x,t) = e^{-\beta t}w(x,t)$ into all terms of the PDE and collect terms. That means you will need to compute $u_t$ and $u_{xx}$ using standard calculus differentiation rules.

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