User:Egm4313.s12.team8.lucas/R6.7

=R6.7=

Problem Statement
Find the seperated ODEs for the heat equation

$$\ \frac{\partial u}{\partial t}=\kappa \frac{\partial ^2 u}{\partial x^2}$$

Solution
Assume $$\ u(x,t)=F(x)\cdot G(t)$$

where:

$$\ F(x)=f(x)$$, a function of only x

$$\ G(t)=f(t)$$, a function of only t

Finding first and second partial derivatives of u with restpect to x gives:

$$\ \frac{\partial u}{\partial x}=F'(x) \cdot G(t)$$

$$\ \frac{\partial^2 u}{\partial x^2}=F''(x) \cdot G(t)$$

Finding first and second partial derivates of u with respect to t gives:

$$\ \frac{\partial u}{\partial t}=F(x) \cdot \dot{G}(t)$$

$$\ \frac{\partial^2 u}{\partial ^2t}=F(x) \cdot \ddot{G}(t)$$

Plugging these into the heat equation gives:

$$\ \frac{\partial u}{\partial t}=\kappa \frac{\partial ^2u}{\partial x^2}  =>  F(x) \cdot \dot{G}(t)=\kappa F''(x) \cdot G(t)$$

Combining like terms gives:

$$\ \frac{\dot G(t)}{\kappa G(t)}=\frac{F''(x)}{F(x)}=c$$

where c is a constant

$$\ \frac{\dot G(t)}{\kappa G(t)}=c   => \dot G(t)=c \kappa G(t)  =>  \dot G(t)-c \kappa G(t)=0$$

$$\ \frac{F(x)}{F(x)}=c    => F(x)=cF(x)     =>    F''(x)-cF(x)=0$$

$$\ \dot G(t)-c \kappa G(t)=0$$

$$\ F''(x)-cF(x)=0$$