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## Homework Statement

Working with a fluids problem I have derived a pde in [itex]v(y,t)[/itex]. It does not seem to matter but I'll write the PDE I derived, in case:

[itex]\frac{\partial v}{\partial t}=\upsilon \frac{\partial ^2 v}{\partial y^2}[/itex]

Assuming I know that the similarity solution below will work in solving the pde:

[itex]v(y,t)=F(\xi)[/itex] where [itex]\xi = \frac{y}{\sqrt{t}}[/itex]

I need to simply show that [itex]F(\xi)[/itex] satisfies the ODE [itex]\frac{d^2 F}{d \xi^{2} }+\frac{\xi}{2\upsilon }\frac{dF}{d \xi}=0[/itex]

subject to boundary conditions [itex]F(0)=U[/itex]

[itex]F(\infty)=0[/itex]

([itex]\upsilon[/itex] and [itex]U[/itex] are constants related to the original PDE problem & its boundary conditions)

## The Attempt at a Solution

I don't quite understand what I am supposed to do here. I tried simply solving the ODE, and I get an answer [itex]F=C\frac{2\upsilon}{x}e^{\frac{-x^2}{4\upsilon}}[/itex]

It was just a quick page of scribbling to see the form of the ODEs solution. It might be slightly wrong, but it does not seem to allow me to show the similarity solution satisfies the ODE.

Please help, with some guidance on what to do. I don't have much experience with similarity solutions, but I have read up on how they are actually derived from PDEs. The above question seems to be simpler than actually deriving it. But I'm a bit lost as to where to start.

Thanks.