6.26 First order constant coefficients

In MATH101 slide 3.17, you saw that the general solution for the differential equation

is $y=Ae^{kx}$ where $A$ is a constant. More generally, let us consider the equation $\frac{dy}{dx}-ky=q(x)$ for a function $q(x)$. This is a first-order linear equation, so we can solve it using the method on slides 6.19-20. In this case $p(x)={-k,}$ so the integrating factor $I(x)={e^{-kx}.}$

Multiplying through by $I(x)$, we obtain the integrable differential equation

which we integrate, to obtain $ye^{-kx}=g(x)+c$ where $g^{\prime}(x)=q(x)e^{-kx}$ and $c$ is a constant. Multiplying through by $e^{kx}$, we get the general solution: