General Solution equals Particular Solution plus Complementary Function

Theorem
Consider the linear first order ODE with constant coefficients:
 * $(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$

The general solution to $(1)$ consists of:


 * the particular solution to $(1)$ for which the arbitrary constant is $0$

plus:
 * the complementary function to $(1)$.

Proof
From Solution to Linear First Order ODE with Constant Coefficients, $(1)$ has the general solution:


 * $\displaystyle y = e^{-a x} \int e^{a x} \map Q x \rd x + C e^{-a x}$

Setting $C = 0$ we get:


 * $\displaystyle y = e^{-a x} \int e^{a x} \map Q x \rd x$

which is a particular solution to $(1)$

By definition, the complementary function to $(1)$ is the general solution to the reduced equation:
 * $\dfrac {\d y} {\d x} + a y = 0$

From First Order ODE: $\dfrac {\d y} {\d x} = k y$, that general solution of $(1)$ is:
 * $y = C e^{-a x}$

Hence the result.

Warning
I am beginning to wonder whether the author of the source work from which this result is taken really understands the material he is presenting.