54.7.99 problem 1711 (book 6.120)

Internal problem ID [12948]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1711 (book 6.120)
Date solved : Wednesday, October 01, 2025 at 02:47:13 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime } y-{y^{\prime }}^{2}+\left (a y-1\right ) y^{\prime }-y \left (y+1\right ) \left (b^{2} y^{2}-a^{2}\right )&=0 \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2+(-1+a*y(x))*diff(y(x),x)-y(x)*(1+y(x))*(b^2*y(x)^2-a^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=-(y[x]*(1 + y[x])*(-a^2 + b^2*y[x]^2)) + (-1 + a*y[x])*D[y[x],x] - D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0; 
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") 
ode = Eq((a**2 - b**2*y(x)**2)*(y(x) + 1)*y(x) + (a*y(x) - 1)*Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*y(x)/2 - sqrt(5*a**2*y(x)**2 + 4*a**2*y(x) - 2*a*y(x) - 4*b**2*y(x)**4 - 4*b**2*y(x)**3 + 4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 + Derivative(y(x), x) + 1/2 cannot be solved by the factorable group method