Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 1 / Section B / 5 / a / ii / Solution

On the real domain $y>0$, $M=y(x^2+\ln y)$ and $N=x$ have $M_y=x^2+\ln y+1$ and $N_x=1$, so this differential one-form is not exact. An integrating factor for a differential one-form is $\mu =1/y$, which is nonzero on this domain. Multiplication givesHence the implicit solution is