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**VonNemo19** Hello. I'm having a bit of trouble understanding the rationale behind the assumption that is made in the variation of parameters method for first and second order linear differential equations.

For the first order equations, my book lays out the property of linear equations that the solution $\displaystyle y$ is the sum of the complementary and a particular solution, that is $\displaystyle y=y_c+y_p$. Then the author shows that the homogeneous equation

$\displaystyle \frac{dy}{dx}+P(x)y=0$

is seperable and has as its solution $\displaystyle y_c=ce^{-\int{P(x)}dx}$ and letting $\displaystyle e^{-\int{P(x)}dx}=y_1(x)$ for convenience, we have $\displaystyle y_c=cy_1(x)$.

Now the author states that in order to find the particular solution $\displaystyle y_p$ we use a procedure known as variation of parameters where we assume that $\displaystyle y_p=uy_1$.

So, I have no trouble at all with the derivation from here on. My question is, how is it that this assumption is made. Or, stated in another way, what was the reasoning behind making this assumption. What is it that would make one believe that the particular solution would be simply some function of the independent variable times the complementary solution?