Chapter 8.4, Problem 25P

An equation of the form $y^{\prime }=f(x)y^2+g(x)y+h(x)$ is called a Riccati equation. If we know one particular solution $y_p,$ then the substitution $y=y_p+\frac{1}{z}$ gives a linear first-order equation for z. We can solve this for $z$ and substitute back to find a solution of the $y$ equation containing one arbitrary constant (see Problem 26). Following this method, check the given $y_p,$ and then solve

(a) $y^{\prime }=x{y}^2-\frac{2}{x}y-\frac{1}{x^3},y_p=\frac{1}{x^2}$;

(b) $y^{\prime }=\frac{2}{x}{y}^2+\frac{1}{x}y-2x,y_p=x$;

(c) $y^{\prime }=e^{-x}{y}^2+y-e^x,y_p=e^x$.