Differential equations/Integrating factors

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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.

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 $I(x)=e^{\int {\frac {N_{x}-M_{y}}{M}}dy}$ .
Substitute the integration factor into the equation $I(x)M(x,y)dx+I(x)N(x,y)dy=0$ and solve.