Proof to Modified EE
Proof to Modified Exact Equation MethodΒΆ
given
\[Mdx+Ndy=0\]
try to use the factor \(\mu\) to make the 1st ODE the exact equation
\[
\begin{gather}
\mu Mdx+\mu Ndy=0\\
\downarrow\\
\frac{\partial \mu M}{\partial y}=\frac{\partial \mu N}{\partial x}\\
\downarrow\\
\mu_y M + \mu M_y = \mu_x N + \mu N_x\\
\downarrow\\
\mu = \frac{\mu_x N - \mu_y M}{M_y - N_x}
\end{gather}
\]
assume \(\mu\) and \((M_y-N_x)/M\) is only dependent to \(y\)
\[\therefore\mu_x = 0\]
implies
\[\mu = \frac{M}{N_x-M_y}\frac{d\mu}{dy}\]
using separable varable method
\[\mu(y) = e^{\int {\frac{(N_x-M_y)}{M}dy}}\]
simiar solution in the case that assume \(\mu\) is only dependent to x.