Hi! Can someone please help me with the heat equation, my working out is below. The question is: Solve

$\displaystyle

\frac {du}{dt} = \frac {d^2u}{dx^2}$

with the boundary conditions:

$\displaystyle

0<=x<=L, T>=0, u(x,0) =f(x), u_x(0,t)=u_x(L,t)=0.

$

My working out is:

$\displaystyle

X'' + sX = 0$

So first case is: $\displaystyle s>0$

when $\displaystyle s= 0$ The general Solution of ODE for $\displaystyle x$

$\displaystyle X(x) = A e^{ xs^{\frac {1}{2}}} + Be^{ xs^{\frac {-1}{2}}}$

Then i differentiate it and got

$\displaystyle

X'(x) = A s^{\frac {1}{2}}e^{ xs^{\frac {1}{2}}} - Bs^{\frac {1}{2}}e^{ xs^{\frac {-1}{2}}}$

From the boundary conditions:

$\displaystyle X'(0) = A s^{\frac {1}{2}} - Bs^{\frac {1}{2}}$

$\displaystyle X'(L) = A s^{\frac {1}{2}}e^{ Ls^{\frac {1}{2}}} - Bs^{\frac {1}{2}}e^{ Ls^{\frac {-1}{2}}}$

And this is where i get lost

I tried solving simultaneously but I don't know how to. Can someone please help me? Thank you