Question

Consider the equation ay'' + by' + cy = d, where a, b, c, and d are constants. (a) find all equilibrium, or constant, solutions of this differential equation. (enter your answers as a comma-separated list of equations.)

Answer 1

Rewrite equation so that it is homogeneous . 
$ay'' +by' + cy = 0$

Solve characteristic equation:
$ar^2 +br + c = 0 \\ \\ r = \frac{(-b) \pm \sqrt{b^2 -4ac}}{2a}$ 

The solutions to the homogeneous equation are:
$y = k_1 e^{r_1 t} + k_2 e^{r_2 t}$

Finally you need the constant term in "cy" to equal 'd' to satisfy the particular solution.
$y= k_1 e^{r_1 t} + k_2 e^{r_2 t} + \frac{d}{c}$