Integrating factor

Integrating factor is a method to solve scalar Linear First Order ODEs. It takes advantage of the product rule of derivatives

$(\mu y)'=\mu 'y+\mu y'$

and attempts to move all y terms into the same differential order.

Note that Separation of variables is generally easier when both options are available.

Procedure

Consider a linear first order ODE.

$y'+p(t)y=g(t)$

We multiply bothsides by integrating factor $\mu (t)$ such that the left hand side to be the exact derivative of a product. For this to work, we need

$\mu '=\mu p$

$\mu (t)=C\exp \left[\int pdt\right]$

Applying the integrating factor, we have

${\begin{aligned}\mu (t)y+\mu (t)p(t)y&=\mu (t)g(t)\\\mu y'+\mu 'y&=\mu g\\(\mu y)'&=\mu g\end{aligned}}$

From there, simple integration and algebra will solve the equation.

$y(t)={\frac {1}{\mu (t)}}\int \mu gdt+C$