6.27 Homogeneous and inhomogeneous equation

The solution in the previous slide consists of two parts: the first term is $g(x)e^{kx}$ where $g(x)$ is a particular integral of $q(x)e^{-kx}$; the second term is $ce^{kx}$, where $c$ is an arbitrary constant. Note that the second term is just the general solution to the equation $\frac{dy}{dx}-ky=0$!

For an arbitrary first-order linear differential equation $\frac{dy}{dx}+p(x)y=q(x)$, we say that the corresponding homogeneous equation is $\frac{dy}{dx}+p(x)y=0$.

The general solution to the equation can always be written as the sum of a particular integral (PI) and the complementary function (CF), which is the general solution to the corresponding homogeneous equation.