Answer
See the work below.
Work Step by Step
(a) The general form of a second-order homogeneous
linear differential equation with constant coefficients is
$$
a y^{\prime \prime}+b y^{\prime}+c =0,
$$
where $a, b$ and $c$ are constants.
(b) the auxiliary equation is:
$$
a r^{\prime \prime}+b r^{\prime}+c =0.
$$
(c)
If the auxiliary equation has two distinct real roots $r_{1}$ and $r_{2}$, the solution is
$$
y(x)=c_{1} e^{r_{1}x}+c_{2} e^{r_{2}x}
$$
If the roots are real and equal, $r_{1}=r_{2}=r$ , so the solution is
$$
y(x)=(c_{1} +c_{2} x) e^{rx}
$$
If the roots are complex, $r_{1},r_{2}= \alpha \pm\beta $, the solution is
$$
y(x)=e^{\alpha x}(c_{1}\cos (\beta x) +c_{2} \sin (\beta x))
$$
