91Ó°ÊÓ

Show that the change of variables \(V=x y\) transforms the differential equation $$ \frac{d y}{d x}=\frac{y}{x} F(x y) $$ into the separable differential equation $$ \frac{1}{V[F(V)+1]} \frac{d V}{d x}=\frac{1}{x} $$

A differential equation of the form $$ y^{\prime}+p(x) y+q(x) y^{2}=r(x) $$ is called a Riccati equation. (a) If \(y=Y(x)\) is a known solution to Equation \((1.8 .22),\) show that the substitution $$ y=Y(x)+v^{-1}(x) $$ reduces it to the linear equation $$ v^{\prime}-[p(x)+2 Y(x) q(x)] v=q(x) $$ (b) Find the general solution to the Riccati equation $$ x^{2} y^{\prime}-x y-x^{2} y^{2}=1, \quad x>0 $$ given that \(y=-x^{-1}\) is a solution.