The Integrating Factor Method of Solving First Order Differential Equations
The differential equationsis separable We can write this equation asIntegrating both sides gives
Some first order differential equations are not separable. Often the most suitable way to solve it is the integrating factor method, which can be used to solve equations of the form
If we multiply both sides by the integrating factor,the equation becomeswe can write this as
Integrating givesand dividing bygivesThe constantcan be found to solve the initial differential equation if we have simultaneous values ofand(if we don't have these then we just incliude the integrating constant C – this is the feneral solution).
Example: Solve the differential equationif
Multiply by the integrating factor to givewhich we can write asWe can integrate this. Obtainingthen dividing bygives
whensothen the solution is
There is no error on the second line. You can divide by x as long as x is not zero. You cannot divide by dx though.