Definition:Complementary Function of Linear First Order ODE With Constant Coefficients

Definition
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 the reduced equation:
 * $\dfrac {\d y} {\d x} + a y = 0$

is the complementary function of $(1)$.

From First Order ODE: $\dfrac {\d y} {\d x} = k y$, the complementary function of $(1)$ is $C e^{-a x}$.