Differential equations/Integrating factors

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Definition

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

${\displaystyle I(x)M(x,y)dx+I(x)N(x,y)dy=0}$

is exact.

Solution

There are 2 approaches to a solution.

1. If the function is of the form ${\displaystyle {\frac {dy}{dx}}+p(x)y=r(x)}$ , then the integrating factor is ${\displaystyle I(x)=e^{\int p(x)dx}}$.

OR

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