55.21.2 problem 2

Internal problem ID [13529]
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 : 2
Date solved : Wednesday, October 01, 2025 at 03:59:24 PM
CAS classification : [_Riccati]

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

✗ Maple

ode:=diff(y(x),x) = y(x)^2+1/x^4*f(1/x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2+1/x^4*f[1/x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

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