# How to Use the Reduction of Order Method for Differential Equations

If you have a differential equation without $y$, but also only derivatives of $y$, then you can use substitution to reduce the order of the equation. You put $u={y}^{\prime }$, which gives you ${u}^{\prime }={y}^{″}$ and ${u}^{″}={y}^{‴}$. This gives you a lower-order differential equation, which you can then solve using the usual methods.

Example 1

Solve the differential equation ${y}^{‴}-7{y}^{″}+12=0$

Let ${y}^{\prime }=u$ and substitute:

 ${u}^{″}-7{u}^{\prime }+12=0.$

This has the characteristic equation $\begin{array}{llll}\hfill {r}^{2}-7r+12& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \left(r-4\right)\left(r-3\right)& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

which has the solutions ${r}_{1}=4$ and ${r}_{2}=3$. Enter ${r}_{1}$ and ${r}_{2}$ into the formula for the solution of the characteristic equation and get

 $u\left(x\right)={C}_{1}{e}^{4x}+{C}_{2}{e}^{3x}$

You now need to substitute back ${y}^{\prime }=u$:

 ${y}^{\prime }={C}_{1}{e}^{4x}+{C}_{2}{e}^{3x}$

By integrating, you get $\begin{array}{llll}\hfill y& =\int {y}^{\prime }\phantom{\rule{0.17em}{0ex}}dx\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\int {C}_{1}{e}^{4x}+{C}_{2}{e}^{3x}\phantom{\rule{0.17em}{0ex}}dx\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{{C}_{1}}{4}{e}^{4x}+\frac{{C}_{2}}{3}{e}^{3x}+{C}_{3}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$