55.21.5 problem 5

Internal problem ID [13532]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.9. Some Transformations
Problem number : 5
Date solved : Wednesday, October 01, 2025 at 04:01:19 PM
CAS classification : [_Riccati]

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

✗ Maple

ode:=x^2*diff(y(x),x) = x^4*y(x)^2+x^(2*n)*f(a*x^n+b)-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^4*y[x]^2+x^(2*n)*f[a*x^n+b]+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") 
n = symbols("n") 
y = Function("y") 
f = Function("f") 
ode = Eq(n**2/4 - x**4*y(x)**2 + x**2*Derivative(y(x), x) - x**(2*n)*f(a*x**n + b) - 1/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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