55.21.11 problem 11

Internal problem ID [13538]
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 : 11
Date solved : Wednesday, October 01, 2025 at 04:19:45 PM
CAS classification : [_Riccati]

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

✗ Maple

ode:=x^2*diff(y(x),x) = x^2*y(x)^2+f(a*ln(x)+b)+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+f[a*Log[x]+b]+1/4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

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