- #1

Gregg

- 459

- 0

## Homework Statement

F(x,t) satisfies

[tex] \frac{\partial^2 F(x,t)}{\partial x^2}=\frac{\partial F(x,t)}{\partial t} [/tex]

With the following boundary conditions

[tex] \frac{\partial F(0,t)}{\partial x}=\frac{\partial F(1,t)}{\partial x} [/tex]

[tex]F(0,t) = F(1,t) [/tex]

[tex] F(x,0) = x^2 \text{ for } x \in (0,1) [/tex]

## The Attempt at a Solution

I've assumed it's separation of variables but I am unsure if I can do this

If it is

[tex] F(x,t) = \phi(x) G(t) [/tex]

I end up with

[tex] \frac{d^2\phi}{d x^2} +\lambda \phi = 0 [/tex]

[tex] \frac{dG}{dt} = -\lambda G [/tex]

So

[tex] \phi(x) = c_1 \cos(x\sqrt{\lambda}) + c_2 \sin(x\sqrt{\lambda}) [/tex]

I also got that the constants are

[tex] c_1 = -\frac{1+\cos(\sqrt{\lambda})}{\sin(\sqrt{\lambda})} [/tex]

[tex] c_2 = \frac{1+\cos(\sqrt{\lambda})}{\cos(\sqrt{\lambda})-1} [/tex]

unsure where to go from here in terms of the function phi. lambda denotes eigenfunctions?