Heat equation - theta function?

[Gekko]

du/dt=d2u/dx2

Show that u(x,t)=(t^a) * theta(xi) where xi=x/sqrt(t) and a is a constant, then theta(xi) satisfys the ODE

a*theta - 0.5 * xi * dtheta/dxi = d2theta/dxi2

Not sure how to start this. Any help most appreciated

(sorry if question isn't easy to ready)

 

[HallsofIvy]

Have you tried the obvious? Put u(x,t)=(t^a) * theta(xi) into the given equation. Since theta(xi) is the only unknow function the result will be an equation in theta.

 

[Gekko]

theta(xi) = u(x,t) / t^a

dtheta/dxi = 1 since the right hand side is just a constant wrt xi so this can't be the approach because it won't satisfy the ODE right?

 

[vela]

If the RHS were a constant, the derivative would be 0, not 1. But it's not constant, so it's not relevant.

The idea is to plug in your expression for u(x,t) into the differential equation:

$$\frac{\partial}{\partial t}[t^a \Theta(\xi(x,t))] = \frac{\partial^2}{\partial x^2}[t^a \Theta(\xi(x,t))]$$

where $\xi(x,t)=x/\sqrt{t}$. You'll need to use the chain rule to express the derivatives with respect to t and x in terms of the derivative with respect to ξ.