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

where 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 is the sum of the particular solution which is and the solution to the homogeneous equation, which is . Combining these, we arrive at the general solution

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?