55.2.50 problem 50

Internal problem ID [13276]
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 : 50
Date solved : Wednesday, October 01, 2025 at 05:02:31 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 443

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

\[ y = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-a b +2 \beta c +\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_1 \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_1 +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (a x +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 c \left (x \alpha +\frac {\beta }{2}\right )\right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_1 +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) c x} \]

✓ Mathematica. Time used: 3.462 (sec). Leaf size: 1312

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

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

✗ Sympy

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

NotImplementedError : The given ODE -Alpha - BETA/x - Gamma/x**2 - a*y(x) - b*y(x)/x - c*y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method