Second-order linear differential equation

A second-order linear differential equation has the form
$$p(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x)y = G(x).$$
If $G(x)=0$, for all $x$, such equations are called homogeneous linear equations.
If $G(x)\neq 0$, the equation is called nonhomogeneous.

For homogeneous and constant coefficient equations,
$$ay''+by'+cy=0,$$
the solution is shown below.

Assume the solution has this form: $e^{rx}$. We will get
$$ar^2+br+c=0,$$
$$\Delta=b^2-4ac,$$
$$r_1=\frac{-b+\sqrt{\Delta}}{2a},$$
$$r_2=\frac{-b-\sqrt{\Delta}}{2a}.$$
For $\Delta>0$,   the solution is                     $c_1e^{r_1x}+c_2e^{r_2x}$.
       $\Delta=0$,   $r_1=r_2$                          $c_1e^{r_1x}+c_2xe^{r_2x}$.
       $\Delta<0$,   $r=\alpha\pm i\beta$         $e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$.

Reference:
https://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf