Question

If two solutions to a partial differential equation are known Psi_1 and Psi_2, then the joint solution Psi=Psi_1+Psi_2 is also a valid solution:

Show that if a third solution (Psi_3 is known) then the joint solution Psi = Psi_1+Psi_2+Psi_3 is also a valid solution

Answer 1

does the superposition principle apply to pdes?

Answer 2

@james

Answer 3

This holds if you have a linear homogeneous equation, yes.

Call these solutions p1, p2 and p3.

Then write
\[ p1 + p2 + p3 = (p1 + p2) + p3 \]
and now use the result for two solutions.

Answer 4

K, that's what i was thinking to do, i just wasn't sure if it was legitimate.

Answer 5

More generally, let
\[ p_j , \ j=1, 2, ..., n \]
be solutions of a linear homogeneous PDE (or ODE). Then
\[ p = \sum_{i=1}^n c_jp_j \]
for any constants \( c_j \) is also a solution.

This is an example of what pre-algebra mentioned: the superposition principle. We can 'superimpose' one solution on another, and the resulting function is still a solution.

This a major advantage of linear equations and reflects physical reality, where waves superimpose themselves all the time, such as sound and light.