Approach of Variation of parameters (Variation of Constants) i have a question about the approach "Variation of parameters (also known as variation of constants).
Imagine we have non-homogene ODE of the form:
$$ y' = a(x) \cdot y + b(x)$$
The homogene solution is found by the Eigenvalue approach or others like seperation of variables.
This leads us to:
$$ y_H = C \cdot \underbrace{e^{\int a(x) \mathop{dx}}}_{y_h} \qquad C \in \mathbb{C}$$
The next step to get the non-homogene solution is to use variation of constants with the approach:
$$ y_P = C(x)\cdot y_h $$
Why do we expect the special solution $y_P$ to be of the same kind as $y_H$? How do I know that the soluation must have the structure $y(x) = y_H(x) + y_P(x)$
I would greatly appreciate answers to the question.
 A: Those are two rather unrelated questions.
For the first one, it's simply because someone figured out that that substitution leads to an equation of the form $C'(x) = \cdots$, which can be solved just by integrating it. I don't know who exactly was first for ODEs of order one, but the general method of variation of constants is credited to Euler and Lagrange.
(By the way, $y_P = C \cdot y_H$ doesn't need to look anything like $y_H$, since $C$ can be arbitrary complicated, so it's a bit of a stretch in general to say that they are “of the same kind”. But in simple cases, $C$ will be a simple function. It's not very far-fetched that if you have, for example, $b(x)=\sin x$ on the right-hand side, and $a(x)$ doesn't contain any trig functions, then you expect $y_P$ to be “some expression containing $\sin x$ and/or $\cos x$”, since how would otherwise that $\sin x$ appear when combining $y_P$ and $y_P'$?)
The second one (regarding $y_H+y_P$) is as very standard fact about linear equations (not only linear differential equations) which is explained in every textbook and surely many times on this site already; see here for one question about the case of second-order linear ODEs.
