The Riccati Equation

Find the general integral of the equation $x^{2}y'=x^{2}y^{2}+xy+1$, which has a particular integral of the form $\dfrac {a}{x}$.

Find the general integral of the equation $y'+y^{2}=2x^{-2}$, which has a particular integral of the form $\dfrac {a}{x}$.

Find the general integral of the equation $4y'+y^{2}+4x^{-2}=0$, which has a particular integral of the form $\dfrac {a}{x}$.

Find the general integral of the equation $x^{2}y'+(xy-2)^{2}=0$, which has a particular integral of the form $\dfrac {a}{x}$.

Solve the equation $xy'+3y+y^{2}=x^{2}$.

Solve the equation $xy'-5y-y^{2}=x^{2}$.

Solve the equation $3xy'-9y-y^{2}=x^{\frac{2}{3}}$.

Solve the equation $5y'+y^{2}=x^{-\frac{12}{5}}$.

Solve the equation $y'+y^{2}=x^{-4}$.

Solve the equation $y'+y^{2}=-x^{-4}$.

Solve the equation $y'-y^{2}=x^{-\frac{8}{3}}$.

Solve the equation $y'-y^{2}=-x^{-\frac{4}{3}}$.