6 The integrating factor

The equation

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

is not exact. However, if we multiply it by $x$ we obtain the equation

$x^3\frac{dy}{dx}+3x^{2}y=x^4.$

This can be re-written as

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

which is an exact equation with solution

$x^{3}y=\int \; <!--nolimits\; -->x^{4}dx$

$\text{so}{x}^{3}y=\frac{1}{5}{x}^5+C$

and hence

$y=\frac{1}{5}{x}^2+\frac{C}{x^3}.$

The function by which we multiplied the given differential equation in order to make it exact is called an integrating factor . In this example the integrating factor is simply $x$ .

Task!

Which of the following differential equations can be made exact by multiplying by $x^2$ ?

1. $\frac{dy}{dx}+\frac{2}{x}y=4$
2. $x\frac{dy}{dx}+3y=x^2$
3. $\frac{1}{x}\frac{dy}{dx}-\frac{1}{x^2}y=x$
   
   
4. $\frac{1}{x}\frac{dy}{dx}+\frac{1}{x^2}y=3.$

   Where possible, write the exact equation in the form $\frac{d}{dx}\left(f\left(x\right)y\right)=g\left(x\right)$ .

Answer