### 3 Exact equations

Consider the differential equation

$\phantom{\rule{2em}{0ex}}\frac{dy}{dx}=3{x}^{2}$

By direct integration we find that the general solution of this equation is

$\phantom{\rule{2em}{0ex}}y={x}^{3}+C$

where $C$ is, as usual, an arbitrary constant of integration.

Next, consider the differential equation

$\phantom{\rule{2em}{0ex}}\frac{d}{dx}\left(yx\right)=3{x}^{2}.$

Again, by direct integration we find that the general solution is

$\phantom{\rule{2em}{0ex}}yx={x}^{3}+C.$

We now divide this equation by $x$ to obtain

$\phantom{\rule{2em}{0ex}}y={x}^{2}+\frac{C}{x}.$

The differential equation $\frac{d}{dx}\left(yx\right)=3{x}^{2}$ is called an exact equation . It can effectively be solved by integrating both sides.

Solve the equations

1. $\frac{dy}{dx}=5{x}^{4}$
2.   $\frac{d}{dx}\left({x}^{3}y\right)=5{x}^{4}$
1. $y={x}^{5}+C$
2. ${x}^{3}y={x}^{5}+C$ so that $y={x}^{2}+\frac{C}{{x}^{3}}.$

If we consider examples of this kind in a more general setting we obtain the following Key Point:

##### Key Point 2

The solution of the equation

$\frac{d}{dx}\left(f\left(x\right)\cdot y\right)=g\left(x\right)$
is
$f\left(x\right)\cdot y=\int g\left(x\right)\phantom{\rule{0.3em}{0ex}}dx\phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}y=\frac{1}{f\left(x\right)}\int g\left(x\right)\phantom{\rule{0.3em}{0ex}}dx$