61.21.1 problem 1

Internal problem ID [12233]
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 : 1
Date solved : Monday, March 31, 2025 at 04:40:07 AM
CAS classification : [_Riccati]

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

✗ Maple

ode:=diff(y(x),x) = y(x)^2+a^2*f(a*x+b); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==y[x]^2+a^2*f[a*x+b]; 
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(-a**2*f(a*x + b) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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