Reduction of order

Summary

Reduction of order is a method to find a second solution of a homogeneous linear equation from a known one. Let $p, q$ be continuous on an open interval $I$, and let $y_1$ be a solution of $y'' + p(x) y' + q(x) y = 0$ with no zero on $I$. Put $y = v\, y_1$. Then $y$ is a solution if and only if $y_1 v'' + (2 y_1' + p\, y_1) v' = 0$, a first-order linear equation for $v'$. One solution gives $y_2 = y_1 \int \frac{e^{-\int p\, dx}}{y_1^2}\, dx$, and $(y_1, y_2)$ is a fundamental system on $I$.

Remarks

Example: $y_1 = e^{x}$ solves $y'' - 2 y' + y = 0$. Here $p = -2$, so $v' = e^{2x}/e^{2x} = 1$, and $y_2 = x e^x$. The Wronskian of $y_1$ and $y_2$ is $e^{-\int p\, dx}$, which has no zero. This is consistent with Abel's identity. The same idea lowers the order by one for a linear equation of order $n$.

Explanations

L1 human

If you know one solution, look for a second one as the first one times an unknown function $v$. The terms without derivatives of $v$ cancel, because $y_1$ is a solution. What remains is a first-order equation for $v'$, which is easy to solve.