Internal problem ID [9639]
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]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha }=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 563
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)
\[ y \left (x \right ) = -\frac {b \left (-\frac {\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {a +\beta }{a}} \left (a x -\sqrt {-b a}\right ) \operatorname {HeunC}\left (0, \frac {-a -\beta }{a}, \frac {2 a +\beta }{2 a}, 0, \frac {2 a^{2}+2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )}{2}+\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {a +\beta }{a}} \left (a x +\sqrt {-b a}\right ) \operatorname {HeunCPrime}\left (0, \frac {-a -\beta }{a}, \frac {2 a +\beta }{2 a}, 0, \frac {2 a^{2}+2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )+\left (\left (\sqrt {-b a}\, x a +a^{2} x^{2}+\left (\beta \,x^{2}+2 b \right ) a +b \beta \right ) \operatorname {HeunC}\left (0, \frac {a +\beta }{a}, \frac {2 a +\beta }{2 a}, 0, \frac {2 a^{2}+2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )+2 a \operatorname {HeunCPrime}\left (0, \frac {a +\beta }{a}, \frac {2 a +\beta }{2 a}, 0, \frac {2 a^{2}+2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right ) \left (-\sqrt {-b a}\, x +b \right )\right ) c_{1} a \right )}{\alpha \left (\frac {\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {a +\beta }{a}} \left (\sqrt {-b a}\, x +b \right ) \operatorname {HeunC}\left (0, \frac {-a -\beta }{a}, \frac {2 a +\beta }{2 a}, 0, \frac {2 a^{2}+2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )}{2}+\operatorname {HeunC}\left (0, \frac {a +\beta }{a}, \frac {2 a +\beta }{2 a}, 0, \frac {2 a^{2}+2 a \beta +\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right ) c_{1} \left (a^{2} b x +\left (-b a \right )^{\frac {3}{2}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.671 (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*}