55.22.76 problem 76

Internal problem ID [13617]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.1-2. Solvable equations and their solutions
Problem number : 76
Date solved : Wednesday, October 01, 2025 at 07:39:55 PM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=a^{2} f^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )-\frac {\left (f \left (x \right )+b \right )^{2} f^{\prime \prime }\left (x \right )}{{f^{\prime }\left (x \right )}^{3}} \end{align*}

✗ Maple

ode:=y(x)*diff(y(x),x)-y(x) = a^2*diff(f(x),x)*diff(diff(f(x),x),x)-(f(x)+b)^2/diff(f(x),x)^3*diff(diff(f(x),x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==a^2*D[ f[x],x]*D[ f[x],{x,2}]-(f[x]+b)^2/( (D[ f[x],x])^3)*D[ f[x],{x,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") 
y = Function("y") 
f = Function("f") 
ode = Eq(-a**2*Derivative(f(x), x)*Derivative(f(x), (x, 2)) + (b + f(x))**2*Derivative(f(x), (x, 2))/Derivative(f(x), x)**3 + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out