The following question was given to us in an exam:
If $0=M dx + N dy$ is an exact equation, in addition to the fact that $\frac{M}{N} = f\Big(\frac{y}{x}\Big)$ is homogeneous, then
$xM_x + yM_y = (xN_x + yN_y)f$.
Now I had absolutely no idea how to prove this question. I tried doing $M = Nf$ and taking derivatives and multiplying by $x$ or $y$, and you get the required R.H.S. but with the extra term $N(\frac{-f_x}{x} + \frac{f_y}{x})$ added. How does one approach a question like that??
I have never encountered a question like that, not even when solving for different types of integrating factors to get an exact equation or when working with a homogeneous equation.
Anyone got any ideas? Please don't post a complete solution.
Thanks.
Best Answer
You had the right idea. If $M(x,y) = N(x,y) F(x,y)$, you have $x M_x + y M_y = x (N F)_x + y (N F)_y = (x N_x + y N_y) F + N (x F_x + y F_y)$. Now if $F(x,y) = f(y/x)$, you have $F_x = - f'(y/x) \frac{y}{x^2}$ and $F_y = \frac{f'(y/x)}{x}$, so $x F_x + y F_y = 0$. Note that the statement that the equation is exact was a red herring.