Toronto Math Forum
MAT2442014F => MAT244 MathLectures => Topic started by: Victor Ivrii 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. http://www.math.toronto.edu/courses/mat244h1/20149/MAT244LN1.html (http://www.math.toronto.edu/courses/mat244h1/20149/MAT244LN1.html)
If equation is exact we try to make it exact by multiplication by $\mu(x,y)$. Then $(\mu M)_y  (\mu N)_x= \mu (M_yN_x) + \mu_y M\mu_x N=0$ 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) $(M_yN_x)/N=f(x)$; then we are looking for $\mu=\mu(x)$ which satisfies $\mu'/\mu = (M_yN_x)/N=f(x)$;
(ii) $(M_yN_x)/M=f(y)$; then we are looking for $\mu=\mu(y)$ which satisfies $\mu'/\mu = (M_yN_x)/M=f(y)$;
(iii) $(M_yN_x)/(x My N)=f(xy)$; then we are looking for $\mu=\mu(xy)$ which satisfies $\mu'/\mu = (M_yN_x)/(x MyN)= f(xy)$.
These three cases are covered in http://www.math.toronto.edu/courses/mat244h1/20149/MAT244LN2.html (http://www.math.toronto.edu/courses/mat244h1/20149/MAT244LN2.html)