Author Topic: Integrating factor  (Read 814 times)

Victor Ivrii

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Integrating factor
« on: October 12, 2014, 08:44:53 PM »
Integrating factor problems will be present in MT and Final. What one needs to know?

Consider $M(x,y)dx+N(x,y)dy=0$. It is exact iff $M_y=N_x$ (where we use notation $M_y=\partial_y M$, and so on).) Actually this is true only for simple connected domains, for other domains this condition is necessary but not sufficient.

If equation is exact we try to make it exact by multiplication by $\mu(x,y)$. Then $(\mu M)_y - (\mu N)_x= \mu (M_y-N_x) + \mu_y M-\mu_x N=0$ is a linear first order PDE (partial derivatives equation) which generally is no more simple than the original ODE.

Still, there are three cases when this PDE could be solved:

(i) $(M_y-N_x)/N=f(x)$; then we are looking for $\mu=\mu(x)$ which satisfies $\mu'/\mu = (M_y-N_x)/N=f(x)$;

(ii) $(M_y-N_x)/M=f(y)$; then we are looking for $\mu=\mu(y)$ which satisfies $\mu'/\mu =- (M_y-N_x)/M=-f(y)$;

(iii)  $(M_y-N_x)/(x M-y N)=f(xy)$; then we are looking for $\mu=\mu(xy)$ which satisfies $\mu'/\mu =- (M_y-N_x)/(x M-yN)= -f(xy)$.

These three cases are covered in
« Last Edit: October 13, 2014, 07:09:07 AM by Victor Ivrii »