Integrating Factors

Definition

Let $Pdx+Qdy=0$ be an ODE.

If there exist a function $F$ s.t. $FPdx+FQdy=0$, then $F(x,y)$ is called an integrating factor

Finding

$Pdx+Qdy=0\Rightarrow FPdx+FQdy=0\Rightarrow \frac{\partial }{\partial y}FP=\frac{\partial }{\partial x}FQ\Rightarrow F_{y}P+F{P}_y=F_{x}Q+F{Q}_x$

If $F=F(x)$ and $F_y=0$, then $F{P}_y=F_{x}Q+F{Q}_x\Rightarrow \frac{dF}{dx}Q=F(P_y-Q_x)\Rightarrow \frac{dF}{dx}\frac{1}{F}=\frac{1}{Q}(P_y-Q_x)=R(x)$. If $R$ depends only on $x$, $\ln |F|=\int Rdx+C\Rightarrow F(x)=\exp (\int R(x)dx)$

If $F=F(y)$ and $F_x=0$, similarly, $\frac{1}{P}(Q_x-P_y)=R^{\ast }(y)$. If $R^{\ast }$ depends only on $y$, $F^{\ast }(y)=\exp (\int R^{\ast }(y)dy)$

Examples

&e^{x} + (e^{x}\cot(y) + 2y \csc(y))y' = 0\\ &\underbrace{e^{x}}_{P}dx + \underbrace{(e^{x}\cot(y) + 2y \csc(y))}_{Q}dy = 0\\ &P_{y}=0,\quad Q_{x}=e^{x}\cot(y), \quad R(x) = \frac{Q_{x}-P_{y}}{P}=\cot(y), F(y)=\exp(\int\cot(y)dy)=\sin(y)\\ &\underbrace{\sin(y)e^{x}}_{P'}dx + \underbrace{(e^{x}\cos(y) + 2y)}_{Q'}dy = 0\\ &P'_{y} = Q'_{x} = \cos(y)e^{x}: \text{Exact ODE}\\ &\int p' dx = \sin(y)e^{x} + h(y) = f\\ &f_{y} = e^{x}\cos(y) + h'(y) = (e^{x}\cos(y) + 2y) \Rightarrow h'(y) = 2y \Rightarrow h(y)=y^{2}\\ &f = \sin(y)e^{x} + y^{2}= C \end{aligned}$$