The subject of Chapter 15 are solutions to second-order homogeneous linear differential equations with constant coefficients, i.e. differential equations of the form
\[
a \frac{d^2 y}{d x^2} + b \frac{d y}{d x} + c y = 0,
\]
where \(a,b,c \in \mathbb{R}\) are constants such that \(a \neq 0\). Then the characteristic polynomial for this equation is given as
\[
a r^2 + br + c = 0.
\]
There are three cases:

1. There are two distinct real roots \(r_1\) and \(r_2\), i.e. \(a r^2 + br + c = (r - r_1) (r-r_2)\). Then \[\{e^{r_1x}, e^{r_2x}\}\] is a fundamental set of solutions.
   
   
2. There is only one real root \(r_1\), i.e. \(a r^2 + br + c = (r - r_1)^2\). Then \[\{e^{r_1x},xe^{r_1x}\}\] is a fundamental set of solutions.
   
   
3. There is a pair of conjugate complex roots \(r_1 = \lambda + i \omega\) and \(r_2 = \lambda - i \omega\). Then both \[\{e^{(\lambda + i \omega)x}, e^{(\lambda - i \omega)x}\}\] and \[\{e^{\lambda x} \cos(\omega x), e^{\lambda x} \sin(\omega x)\}\] are fundamental sets of solutions. 

Example: Consider 
\[
\frac{d^2 y}{d x^2} - y = 0.
\]
Then \(r^2 - 1 = 0\) splits as \((r-1) (r+1)\), hence we are in the first case. We get the fundamental set of solutions \(\{e^{x}, e^{-x}\}\) we already got on the reminder sheet for Chapter 13.