is a solution of the partial differential equation

and boundary conditions

C'(0,t) = 0
C'(L,t) = 0

First,

Next, consider

Comparing these two expressions shows that the function satisfies the Partial Differential Equation. Next, note that

Since the sine term equals the sine of zero, which is 0.

Since the sine term equals the sine of n times pi, which is 0.