55.2.54 problem 54

Internal problem ID [13280]
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 : 54
Date solved : Wednesday, October 01, 2025 at 05:17:38 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 215200

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 1.909 (sec). Leaf size: 2649

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

Too large to display

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

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