Show whether the following PDEs satisfy the principle of sup

Show whether the following PDEs satisfy the principle of superposition for two solutions and explain why or why not.

Solution

Let u1 be a solution of the linear PDE
     L[u] = f1

let u2 be a solution of the linear PDE

        L[u] = f2.

Then, for any any constants c1 and c2, c1u1 + c2u2 is a solution of

L[u] = c1f1 + c2f2. T

That is, L[c1u1 + c2u2] = c1f1 + c2f2. (2)

In particular, when f1 = 0 and f2 = 0, (2) implies that if u1 and u2 are solutions of the homogeneous linear PDE L[u] = 0, then c1u1 + c2u2 will also be a solution of L[u] = 0