Paper 2, Section I, B

Consider the ordinary differential equation

$P(x, y)+Q(x, y) \frac{d y}{d x}=0 .$

State an equation to be satisfied by $P$ and $Q$ that ensures that equation $(*)$ is exact. In this case, express the general solution of equation $(*)$ in terms of a function $F(x, y)$ which should be defined in terms of $P$ and $Q$.

Consider the equation

$\frac{d y}{d x}=-\frac{4 x+3 y}{3 x+3 y^{2}}$

satisfying the boundary condition $y(1)=2$. Find an explicit relation between $y$ and $x$.

*Typos? Please submit corrections to this page on GitHub.*