Now we apply the simplification of fixing the temperature so that the temperature at x@0 and x@2L are equal. This can be achieved with two heat sources.

`> `
**restart;**

Start with the general equation as always.

`> `
**Theta:=x->M*exp(-m*x)+N*exp(m*x);**

Set the equations for the endpoints, and equate them to solve for the M and N.

`> `
**eqn1:=Theta[0]=Theta(0);
eqn2:=Theta[0]=Theta(2*L);**

`> `
**s:=solve({eqn1,eqn2},{M,N});**

`> `
**assign(s);**

`> `
**theta:=simplify(Theta(x)):
Theta:=unapply(theta,x):**

This is our general equation with the constants solved for.

`> `
**Theta(x);**

The temperature at the midpoint.

`> `
**Theta(L);**

The temperature minimum (@L) can be reduced to a simple equation.

`> `
**bookans:=Theta[0]*(2/(exp(-m*L)+exp(m*L)));**

`> `
**convert(bookans,trig);**

`> `
**difference:=simplify(Theta(L)-bookans);**

Heat Conduction in the Steady State