**PDE: Separation of Variables**

1) Let .

2) Note that , , , , .

3) Rearrange the equation such that LHS is a function of only, RHS is a function of only.

4) Thus, LHS=RHS=some constant .

5) Solve the two separate ODEs.

**Wave Equation**

where , , .

**Solution of Wave Equation (with Fourier sine coefficients)**

where

**d’Alembert’s solution of Wave Equation**

**Heat Equation**

, .

**Solution of Heat Equation**

where are Fourier sine coefficients of .