61.2.57 problem 57

Internal problem ID [12063]
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 : 57
Date solved : Tuesday, January 28, 2025 at 12:03:07 AM
CAS classification : [_rational, _Riccati]

\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma &=0 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 858

dsolve((a*x^2+b)*diff(y(x),x)+alpha*y(x)^2+beta*x*y(x)+gamma=0,y(x), singsol=all)
 

\[ y = -\frac {\left (a x -\sqrt {-a b}\right )^{2} \gamma \left (-2 \left (\frac {a x -\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \operatorname {hypergeom}\left (\left [\frac {2 a -\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}+\left (2 a -\beta \right ) b}{2 a b}\right ], \left [\frac {2 a -\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right ) \left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{-\frac {\beta }{2 a}} c_{1} \sqrt {-a b}+\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} \operatorname {hypergeom}\left (\left [\frac {\beta }{2 a}, \frac {b \beta +\sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}}{2 a b}\right ], \left [\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right ) \left (a x +\sqrt {-a b}\right )\right ) b}{2 a \left (-a b \left (-\sqrt {-a b}\, x +b \right ) \left (a \,x^{2}+b \right ) \operatorname {HeunCPrime}\left (0, \frac {-a +\beta }{a}, -\frac {\sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, -\frac {2 \sqrt {-a b}}{a x -\sqrt {-a b}}\right )+\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{-\frac {\beta }{2 a}} c_{1} \left (\frac {a x -\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (\left (-\frac {b}{2}+\frac {a \,x^{2}}{2}-\sqrt {-a b}\, x \right ) \sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}+a \left (b +x^{2} \left (a -\beta \right )\right ) x \sqrt {-a b}+b \left (\left (a -\frac {\beta }{2}\right ) b +a \,x^{2} \left (a -\frac {3 \beta }{2}\right )\right )\right ) b \operatorname {hypergeom}\left (\left [\frac {2 a -\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}+\left (2 a -\beta \right ) b}{2 a b}\right ], \left [\frac {2 a -\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )+2 a \left (\frac {a x -\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} b^{2} c_{1} \left (a \,x^{2}+2 \sqrt {-a b}\, x -b \right ) \operatorname {HeunCPrime}\left (0, \frac {a -\beta }{a}, -\frac {\sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, -\frac {2 \sqrt {-a b}}{a x -\sqrt {-a b}}\right )+\frac {\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} \operatorname {hypergeom}\left (\left [\frac {\beta }{2 a}, \frac {b \beta +\sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}}{2 a b}\right ], \left [\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right ) \left (\sqrt {-a b}\, x +b \right ) \left (-b \beta +\sqrt {4 \alpha \gamma a b +b^{2} \beta ^{2}}\right ) \left (a \,x^{2}+b \right )}{4}\right )} \]

✓ Solution by Mathematica

Time used: 0.636 (sec). Leaf size: 598

DSolve[(a*x^2+b)*D[y[x],x]+\[Alpha]*y[x]^2+\[Beta]*x*y[x]+\[Gamma]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {i \left (c_1 \left (\sqrt {4 a \alpha \gamma +b \beta ^2}-2 a \sqrt {b}-\sqrt {b} \beta \right ) P_{\frac {\beta }{2 a}+1}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+2 i \sqrt {a} x (a+\beta ) Q_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+\left (\sqrt {4 a \alpha \gamma +b \beta ^2}-2 a \sqrt {b}-\sqrt {b} \beta \right ) Q_{\frac {\beta }{2 a}+1}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right )-2 \sqrt {a} c_1 x (a+\beta ) P_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{2 \sqrt {a} \alpha \left (c_1 P_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+Q_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right )} \\ y(x)\to \frac {-2 x (a+\beta )+\frac {i \left (\sqrt {4 a \alpha \gamma +b \beta ^2}-2 a \sqrt {b}-\sqrt {b} \beta \right ) P_{\frac {\beta }{2 a}+1}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} P_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}}{2 \alpha } \\ \end{align*}