Proof to Modified EE

Proof to Modified Exact Equation Method¶

given

M d x + N d y = 0

try to use the factor $μ$ to make the 1^st ODE the exact equation

\begin{matrix} μ M d x + μ N d y = 0 \\ ↓ \\ \frac{\partial μ M}{\partial y} = \frac{\partial μ N}{\partial x} \\ ↓ \\ μ_{y} M + μ M_{y} = μ_{x} N + μ N_{x} \\ ↓ \\ μ = \frac{μ_{x} N - μ_{y} M}{M_{y} - N_{x}} \end{matrix}

assume $μ$ and $(M_{y} - N_{x}) / M$ is only dependent to $y$

∴ μ_{x} = 0

implies

μ = \frac{M}{N_{x} - M_{y}} \frac{d μ}{d y}

μ (y) = e^{\int \frac{(N_{x} - M_{y})}{M} d y}

simiar solution in the case that assume $μ$ is only dependent to x.