Is linear combination of particular solutions solution of DE as well

Question

I know linear combination of homogeneous solutions is always solution of the DE. Is this also true for particular solutions? When $y_1$ and $y_2$ are the particular solutions, is for example $y=5y_1+7y_2$ also a solution?

Answer 1

This is not true. There is ambiguity with the homogenous solution if no initial condition is specified. But the particular solution is a function which solves the inhomogenous equation; there is no ambiguity because when you feed it into the equation it will spit out exactly the function that makes the equation inhomogenous. Try it out for yourself.

Also, the statement that a linear combination of the homogenous solutions will produce another solution is only true for linear equations!