Internal problem ID [9640]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma =0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 682
dsolve((a*x^2+b)*diff(y(x),x)+alpha*y(x)^2+beta*x*y(x)+gamma=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2 b \gamma \left (\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {a -\beta }{a}} \operatorname {HeunC}\left (0, \frac {-a +\beta }{a}, -\frac {\sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )-\operatorname {HeunC}\left (0, \frac {a -\beta }{a}, -\frac {\sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right ) c_{1} \right ) \left (a x -\sqrt {-b a}\right )}{2 c_{1} \left (\frac {\left (-\sqrt {-b a}\, x -b \right ) \sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}}{2}+b \left (\frac {\sqrt {-b a}\, \beta x}{2}+\left (a -\frac {\beta }{2}\right ) b +a \,x^{2} \left (a -\beta \right )\right )\right ) \operatorname {HeunC}\left (0, \frac {a -\beta }{a}, -\frac {\sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )+\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {a -\beta }{a}} \left (\sqrt {-b a}\, x +b \right ) \left (-b \beta +\sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}\right ) \operatorname {HeunC}\left (0, \frac {-a +\beta }{a}, -\frac {\sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )-4 a b \left (-\sqrt {-b a}\, x +b \right ) \left (\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {a -\beta }{a}} \operatorname {HeunCPrime}\left (0, \frac {-a +\beta }{a}, -\frac {\sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )-\operatorname {HeunCPrime}\left (0, \frac {a -\beta }{a}, -\frac {\sqrt {4 \gamma \alpha b a +b^{2} \beta ^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right ) c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 0.667 (sec). Leaf size: 598
DSolve[(a*x^2+b)*y'[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*}