# How to Solve Exact First Order Differential Equations

A differential equation is exact if you can integrate the right-hand side and the left-hand side directly. This is the simplest type of differential equations.

Rule

### ExactDifferentialEquations

 ${y}^{\prime }=f\left(x\right)⇒y=\int f\left(x\right)\phantom{\rule{0.17em}{0ex}}dx+C$

Example 1

Solve the differential equation ${y}^{\prime }=\frac{1}{x}+2x$

$\begin{array}{llll}\hfill y& =\int {y}^{\prime }\phantom{\rule{0.17em}{0ex}}dx\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\int \frac{1}{x}+2x\phantom{\rule{0.17em}{0ex}}dx\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}|x|+{x}^{2}+C\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{lll}\hfill y=\int {y}^{\prime }\phantom{\rule{0.17em}{0ex}}dx=\int \frac{1}{x}+2x\phantom{\rule{0.17em}{0ex}}dx=\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}|x|+{x}^{2}+C& \phantom{\rule{2em}{0ex}}& \hfill \end{array}$