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/MAT244-LN1.htmlIf 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

http://www.math.toronto.edu/courses/mat244h1/20149/MAT244-LN2.html