Higher-order ODE

We have a few tools for solving higher-order ODEs (where the $n$-th derivative is higher than 2), none of which are pretty.

For linear higher-order ODEs, the same process as second-order linear ODEs applies. We just have more terms. So for example, $y^{\prime \prime \prime }+3y^{\prime \prime }-4y=0$ has a characteristic equation with roots $1,-2,-2$. So we have:

$$y=c_1{e}^x+c_2{e}^{-2x}+c_{3}x{e}^{-3x}$$

And similarly for a mix of complex roots. Computing higher-order polynomials tends to be difficult, so we usually go for numerical methods the higher we go.