Sum of homogeneous and inhomogeneous solutions also form a solution

Question

For some linear differential operator, $L$, an inhomogeneous differential equation can be formed: $$ L~y(x) = F(x) \text{ with some solution } y_p (x).$$ Similarly a homogeneous equation could be formed such that: $$ L~y(x) = 0 \text{ with some solution } y_c (x).$$ My question is, how would I go about showing that $y (x) = y_p (x) + y_c (x)$ is also a solution to the inhomogeneous equation?

I could find solutions to both equations, sum them and resubstitute them in but would this be sufficient. As such, I can show that this is correct for some particular form of L but I am not sure how to show this for any general case.

Thanks for any advice given, Sean.

Answer 1

If $L$ is linear then what does that mean?

$$L(a+b)=?.$$

Now apply this to

$$L(y_p+y_c).$$

In fact this only used the fact that $L$ is additive. If $L$ is linear it also has the property

$$L(\lambda a)=\lambda La,$$ for a scalar $\lambda$.

This shows that $y_p+\lambda y_c$ is a solution also.