Hey forum, I've been asking myself this question ever since I first was introduced to differential equations. Suppose we have a nonhomogeneous differential equation of the form

$\displaystyle y'(x) + a \cdot y(x) = f(x) \, ,$

where $\displaystyle a$ is a constant. Exactly WHY is the general solution a sum of the particular solution and the solution to the homogeneous solution? Why? Is there a proof?

E.g. the general solution to $\displaystyle y'(x) + 5 \cdot y(x) = 3 +15x $ is the sum of the particular solution which is $\displaystyle y_p = 3x$ and the solution to the homogeneous equation, which is $\displaystyle y_h = C \cdot e^{-5x}$. Combining these, we arrive at the general solution

$\displaystyle y = y_h + y_p = 3x + C \cdot e^{-5x}$

Sure, you could verify that it satisfies the equation but that doesn't answer my question. What justifies combining different solutions to acquire a general one? Why do we add the particular and homogeneous one? Proof? Any intuitive explanation?