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Glossary
Integrating factors as functions of independent variable
Suppose that an integrating factor depends only on the independent variable only. Then \( \mu = \mu (x) \) satisfies the differential equation
\[
\frac{\partial}{\partial y} \left( \mu \,M(x,y)\right) = \frac{\partial}{\partial x} \left( \mu \,N(x,y)\right)
\qquad \Longleftrightarrow \qquad \mu \, \frac{\partial M(x,y)}{\partial y} = \frac{{\text d} \mu}{{\text d} x} \, N(x,y) + \mu \,
\frac{\partial N(x,y)}{\partial x} .
\]
Then we get the equation
\[
\frac{1}{\mu} \,\frac{{\text d} \mu}{{\text d} x} = \frac{M_y - N_x}{N(x,y)} \qquad \mbox{must be a function of $x$ only}
\]
because the left-hand side is a function of x. In this case, variable are separated and upon integration, we obtain
\[
\mu (x) = \exp \left\{ \int \frac{M_y - N_x}{N(x,y)} \, {\text d} x \right\} .
\]
Example:
PolarPlot[{Exp[Cos[x]] - 2*Cos[4*x], x}, {x, 0, 2*Pi}]
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