55.2.57 problem 57

Internal problem ID [13283]
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 : Wednesday, October 01, 2025 at 05:36:29 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*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 858

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.409 (sec). Leaf size: 598

ode=(a*x^2+b)*D[y[x],x]+\[Alpha]*y[x]^2+\[Beta]*x*y[x]+\[Gamma]==0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
Gamma = symbols("Gamma") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(Alpha*y(x)**2 + BETA*x*y(x) + Gamma + (a*x**2 + b)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out