ODE's for 2 space Heat equation

[beetle2]

Homework Statement

The Heat equation in two space is

$\alpha ^2 \left[\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2} \right]=\frac{\partial u}{\partial t}$

Assuming separation solution of the form $u(x,y,t)=F(x)G(y)H(t)$ find ordinary differential equations satisfied by F,G and H.

Homework Equations

Heat Equation

The Attempt at a Solution

Because we can assume $u(x,y,t)=F(x)G(y)H(t)$

Is the first step that we require that


$\alpha^2\frac{F''}{F}+\alpha^2\frac{G''}{G}=\frac{H'}{H}$

therefore we need to solve


$F''+\frac{k}{\alpha^2}F+G''+\frac{k}{\alpha^2}G=0$

and

$H'-kH=0$


Is this right so far?

 

[ehild]

The Attempt at a Solution

Because we can assume $u(x,y,t)=F(x)G(y)H(t)$

Is the first step that we require that


$\alpha^2\frac{F''}{F}+\alpha^2\frac{G''}{G}=\frac{H'}{H}$

therefore we need to solve


$F''+\frac{k}{\alpha^2}F+G''+\frac{k}{\alpha^2}G=0$

and

$H'-kH=0$


Is this right so far?

No. F"/F is a function of x, F"/F=f(x). Similarly, G"/G=g(y) and H"/H=h(t)

α²(f(x)+g(y)) = h(t) holds only when f, g, h are all constant functions: f(x)=K, g(y)=L, h(t)=M

with the condition that α²(K+L)=M
Can you proceed from here?

ehild

 

[beetle2]

Do I have to try to combine the first functions and set the right side to = 0 ?

My examples in my notes are all of the form

$\alpha^2\frac{F''}{F}=\frac{ G'}{G}$