Integrating Factor for Second Order ODEs

Definition

If there is a function $μ (x, y)$ such that

μ (x, y) M (x, y) d x + μ (x, y) N (x, y) d y = 0

is exact, then $μ (x, y)$ is an integrating factor

Finding Integrating Factor

Let $\frac{\partial f}{\partial x} = f_{x}$ and $\frac{\partial f}{\partial y} = f_{y}$

Solve $μ_{y} M + μ M_{y} = μ_{x} N + μ N_{x}$
- PDE, so generally hard to solve, simplify by assuming $μ (x, y) = μ (x)$ or $μ (x, y) = μ (y)$
If we assume $μ (x, y) = μ (x)$ , equation reduces to

μ M_{y} = μ_{x} N + μ N_{x} ⟹ \frac{d μ}{d x} = μ (x) \frac{M_{y} - N_{x}}{N}

Assume, $μ (x, y) = μ (y)$ , reduces to

μ_{y} M + μ M_{y} = μ N_{x} ⟹ \frac{d μ}{d y} = μ (y) \frac{N_{x} - M_{y}}{M}