Separation of Variables

Separation of variables can be applied to linear partial differential equations, using the principle of superposition starting with the assumption that .

  1. Assume that the solution is and hence write the PDE in terms of , , , and .
  2. Divide through by to isolate and .
  3. Re-arrange to make each side of the equation only dependent on either or .
  4. Given both sides are independent, they must be equal to a constant, (separation constant).
  5. Solve these two equations using separation of variables (in an exam this is the only technique required for simplicity).
  6. Specify a solution given by the boundary condition.

Example One

Then divide by

LHS depends on only, RHS depends on only. Since and are independent this means each side must be equal to a separation constant, so…

The ordinary differential equations are then separable (don’t have to be but will be in exams for the sake of time!)…

Similarly…

Then,

By the principle of superposition, the general solution is given by the sum over all values of and :

To find a specific solution, we use the boundary condition that , such that…

Hence, using the boundary condition,

Or…

So that the solution is: