54.7.139 problem 1755 (book 6.164)

Internal problem ID [12988]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1755 (book 6.164)
Date solved : Wednesday, October 01, 2025 at 02:55:44 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 19

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

\begin{align*} y &= 0 \\ y &= \left (\frac {c_1 x +c_2}{n}\right )^{n} \\ \end{align*}

✓ Mathematica. Time used: 0.357 (sec). Leaf size: 17

ode=(1 - n)*D[y[x],x]^2 + n*y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 (x-c_1 n){}^n \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(n*y(x)*Derivative(y(x), (x, 2)) - (n - 1)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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