### 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.

##### Task!

Solve the equations

- $\frac{dy}{dx}=5{x}^{4}$
- $\frac{d}{dx}\left({x}^{3}y\right)=5{x}^{4}$

- $y={x}^{5}+C$
- ${x}^{3}y={x}^{5}+C$ 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