Characteristic equation

The idea comes from the derivative of \(e^{rt}\) being a multiple of itself. Substituting \(y=e^{rt}\) into \(ay''+by'+cy=0\) gives \(e^{rt}(ar^2+br+c)=0\), and since \(e^{rt}\) never vanishes, it is enough that \(r\) be a root of the polynomial. Each root gives a solution, and by the superposition principle their combinations are solutions too.

The discriminant \(b^2-4ac\) separates three cases. If it is positive, there are two real roots \(r_1\neq r_2\) and \(y=c_1e^{r_1t}+c_2e^{r_2t}\). If it is zero, there is a double root \(r\) and the second solution is \(te^{rt}\), so \(y=(c_1+c_2t)e^{rt}\). If it is negative, the roots are \(\alpha\pm\beta i\) and \(y=e^{\alpha t}(c_1\cos\beta t+c_2\sin\beta t)\).

With \(y''+2y'+5y=0\), \(r^2+2r+5=0\) gives \(r=-1\pm2i\): oscillations of frequency \(2\) that fade like \(e^{-t}\). In every case the two solutions have nonzero Wronskian, so they form a basis. The sign of the real part of the roots decides whether solutions grow or decay. A common mistake is forgetting the double-root case and proposing only \(c_1e^{rt}\).