3 Exact equations

Consider the differential equation

$\frac{dy}{dx}=3x^2$

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

$y=x^3+C$

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

Next, consider the differential equation

$\frac{d}{dx}\left(yx\right)=3x^2.$

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

$yx=x^3+C.$

We now divide this equation by $x$ to obtain

$y=x^2+\frac{C}{x}.$

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

Task!

Solve the equations

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

Answer