### The story so far...

 We have found the general solutions of Laplace's equation, . The general solutions are: Applying the boundary conditions of the hot plate problem leaves as physically sensible solutions. A linear combination of all of these with coefficients bn is .

### Applying the final boundary condition

 Compare the linear combination above with the Fourier sine series: . They are identical for l=10 and y=0 (because then the e-term is 1). The BC for the heated edge hasn't been used yet: T(x,0)=100. Therefore, we can substitute: . The Fourier coefficients are, generally: . So, in this case: . Integrate: , insert limits: , and tidy up: . Finally, insert the bn into T(x,y): .

And __that's it__ !

### Summary - Laplace's equation

• General solution
1. separation of variables: T(x,y)=X(x)·Y(y)