55.2.16 problem 16

Internal problem ID [13242]
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 : 16
Date solved : Wednesday, October 01, 2025 at 03:57:08 AM
CAS classification : [_Riccati]

\begin{align*} x^{2} y^{\prime }&=x^{2} y^{2}+a \,x^{2 m} \left (b \,x^{m}+c \right )^{n}-\frac {n^{2}}{4}+\frac {1}{4} \end{align*}

✗ Maple

ode:=x^2*diff(y(x),x) = x^2*y(x)^2+a*x^(2*m)*(b*x^m+c)^n-1/4*n^2+1/4; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=x^2*D[y[x],x]==x^2*y[x]^2+a*x^(2*m)*(b*x^m+c)^n+1/4*(1-n^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*x**(2*m)*(b*x**m + c)**n + n**2/4 - x**2*y(x)**2 + x**2*Derivative(y(x), x) - 1/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (a*x**(2*m)*(b*x**m + c)**n - n**2/4 + x**2*y(x)**2 + 1/4)/x**2 cannot be solved by the factorable group method