54.7.165 problem 1783 (book 6.192)

Internal problem ID [13014]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1783 (book 6.192)
Date solved : Wednesday, October 01, 2025 at 02:57:08 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (y^{2}+1\right ) y^{\prime \prime }-3 y {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 41

ode:=(1+y(x)^2)*diff(diff(y(x),x),x)-3*y(x)*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i \\ y &= i \\ y &= \sqrt {-\frac {1}{c_1^{2} x^{2}+2 c_1 c_2 x +c_2^{2}-1}}\, \left (c_1 x +c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.763 (sec). Leaf size: 173

ode=-3*y[x]*D[y[x],x]^2 + (1 + y[x]^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i c_1 (x+c_2)}{\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}\\ y(x)&\to \frac {i c_1 (x+c_2)}{\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}\\ y(x)&\to -\frac {i c_1}{\sqrt {c_1{}^2}}\\ y(x)&\to \frac {i c_1}{\sqrt {c_1{}^2}}\\ y(x)&\to -\frac {i (x+c_2)}{\sqrt {(x+c_2){}^2}}\\ y(x)&\to \frac {i (x+c_2)}{\sqrt {(x+c_2){}^2}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x)**2 + 1)*Derivative(y(x), (x, 2)) - 3*y(x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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