60.7.94 problem 1706 (book 6.115)

Internal problem ID [11644]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1706 (book 6.115)
Date solved : Sunday, March 30, 2025 at 08:32:35 PM
CAS classification : [NONE]

\begin{align*} y^{\prime \prime } y-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-f^{\prime }\left (x \right ) y-y^{3}&=0 \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2+f(x)*diff(y(x),x)-diff(f(x),x)*y(x)-y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 36.227 (sec). Leaf size: 197

ode=-y[x]^3 - y[x]*Derivative[1][f][x] + f[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]
 

\begin{align*} y(x)\to -\frac {\exp \left (c_2-\int _1^x\frac {y(K[3])^3+\left (c_1+\int _1^{K[3]}-\frac {y(K[1])^3+f''(K[1]) y(K[1])-f(K[1]) y''(K[1])}{y(K[1])^2}dK[1]\right ){}^2 y(K[3])^2+f''(K[3]) y(K[3])-f(K[3]) y''(K[3])}{y(K[3])^2 \left (c_1+\int _1^{K[3]}-\frac {y(K[1])^3+f''(K[1]) y(K[1])-f(K[1]) y''(K[1])}{y(K[1])^2}dK[1]\right )}dK[3]\right )}{\int _1^x-\frac {y(K[1])^3+f''(K[1]) y(K[1])-f(K[1]) y''(K[1])}{y(K[1])^2}dK[1]+c_1} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
ode = Eq(f(x)*Derivative(y(x), x) - y(x)**3 - y(x)*Derivative(f(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 -sqrt(f(x)**2 - 4*y(x)**3 - 4*y(x)*Derivative(f(x), x) + 4*y(x)*Derivative(y(x), (x, 2)))/2 - f(x)/2 + Derivative(y(x), x) cannot be solved by the factorable group method