Internal problem ID [10386]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 56.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y=-\frac {b \left (a +\beta \right )}{\alpha }} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 517
dsolve((a*x^2+b)*diff(y(x),x)+alpha*y(x)^2+beta*x*y(x)+b/alpha*(a+beta)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {b \,a^{2} \left (-\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}+b \right ) \left (a \,x^{2}+2 \sqrt {-a b}\, x -b \right ) \operatorname {HeunCPrime}\left (0, -1-\frac {\beta }{a}, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )}{2}-2 c_{1} b \left (\left (3 a \,x^{2}-b \right ) \sqrt {-a b}+x a \left (a \,x^{2}-3 b \right )\right ) a \operatorname {HeunCPrime}\left (0, \frac {\beta }{a}+1, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )+\left (a \,x^{2}+b \right ) \left (\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}-2 \sqrt {-a b}\, x -b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+c_{1} \left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} \left (\left (-a^{2} x^{2}+\left (-x^{2} \beta -2 b \right ) a -b \beta \right ) \sqrt {-a b}+a^{2} b x \right )\right )\right )}{\left (-\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \sqrt {-a b}\, \left (a \,x^{2}+b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} a^{2} b c_{1} \left (-\sqrt {-a b}\, x +b \right )\right ) \left (a x -\sqrt {-a b}\right )^{2} \alpha } \]
✓ Solution by Mathematica
Time used: 1.111 (sec). Leaf size: 27
DSolve[(a*x^2+b)*y'[x]+\[Alpha]*y[x]^2+\[Beta]*x*y[x]+b/\[Alpha]*(a+\[Beta])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x (a+\beta )}{\alpha } \\ y(x)\to -\frac {x (a+\beta )}{\alpha } \\ \end{align*}