- #1

- 41

- 2

i am having difficulty trying to find a change of variables to solve this partial differential equation

[tex]\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}[/tex]

not sure how to pluck out a change of variables by looking at the equation as its definately not obvious to the untrained eye (me).

have tried

[tex]f(x,t) = p(\eta) [/tex] (not sure if i am even on the right track doing this)

where

[tex]\eta = \frac{x}{t^{\gamma}} [/tex]

but that just gets me in a giant mess with

[tex]-\gamma\frac{x}{t}\frac{\partial p}{\partial \eta} = \frac{\partial ^2 p}{\partial \eta ^2} [/tex]

i may have done [tex]\frac{\partial ^2 f}{\partial x^2}= \frac{\partial \eta}{\partial x}\frac{\partial ^2 p}{\partial \eta ^2}\frac{\partial \eta}{\partial x} [/tex] wrongly.

any suggestions?

thanks for the time