Differential equations/Integrating factors

Definition

If the expression $M(x,y)dx+N(x,y)dy=0$ is not exact or homogeneous, an integrating factor $I(x)$ can be found so that the equation:

$I(x)M(x,y)dx+I(x)N(x,y)dy=0$

is exact.

Solution

There are 2 approaches to a solution.

1. If the function is of the form $\frac{dy}{dx}+p(x)y=r(x)$ , then the integrating factor is $I(x)=e^{\int p(x)dx}$.
   
   OR
   
   If the function is of the standard form $M(x,y)dx+N(x,y)dy=0$ , then the integrating factor is $I(x)=e^{\int \frac{M_y-N_x}{N}dx}$ or $I(x)=e^{\int \frac{N_x-M_y}{M}dy}$.
2. Substitute the integration factor into the equation $I(x)M(x,y)dx+I(x)N(x,y)dy=0$ and solve.