User:Egm4313.s12.team2.nicellis44

$$\frac{\partial u }{\partial t}=\kappa\frac{\partial^2 u}{\partial x^2} $$ First, decompose u into its component functions in x and t $$ u(x,t)=F(x)G(t)  $$ Then substitute this expression for u back into the heat equation to yield: $$ F(x)*\dot G(t)=\kappa*(\ddot F(x)*G(t)  $$ Rearrange so that X varying terms are on one side of the equation and T varying terms on the other.   $$\frac {\dot G(t)}{\kappa G(t)}=\frac{\kappa \ddot F(x)}{F(x)}=C (constant) $$ As both equations are equal to the same constant C, they can be separated and both independently set equal to C.    $$\frac {\dot G(t)}{\kappa G(t)}=C $$, $$ \frac{\kappa \ddot F(x)}{F(x)}=C $$ These equations are then rearranged to set them equal to zero.  $$ \dot G(t)-C \kappa G(t)=0 $$  ,  $$ \kappa \ddot F(x)-C F(x)=0 $$ The resulting homogenous differential equations can then be represented in standard ODE notation.   $$ (1)\frac{dG} {dt}-(C \kappa) G=0 $$ $$  (\kappa) \frac{d^2 F}{dx^2}-(C)F=0 $$