Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 16 / a / i / Solution

Write the equation as $Pdx+Qdy=0$, with $P=3x+y$ and $Q=2x$. Thenso a suitable integrating factor is $\mu =|x{|}^{-1/2}$ on either half-line. For example, on $x>0$ the resulting exact form isHence the solution, written to cover either sign of $x$, isThe original equation remains meaningful at $x=0$, where it requires $y=0$. Only $C=0$ gives a finite differentiable extension across zero, namely $y=-x$.