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 constant of integration is $0$

plus:

the complementary function to $(1)$.

Proof

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

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

Setting $C = 0$ we get:

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

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.

$\blacksquare$

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.

Sources

• 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.3$ The form of the general solution