54.7.196 problem 1819 (book 6.228)

Internal problem ID [13045]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1819 (book 6.228)
Date solved : Friday, October 03, 2025 at 03:58:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.119 (sec). Leaf size: 79

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_f} -\operatorname {RootOf}\left (-\tan \left (\frac {1}{\textit {\_Z}}\right ) c_1 \textit {\_Z} +\textit {\_f} c_1 \tan \left (\frac {1}{\textit {\_Z}}\right )+\tan \left (\frac {1}{\textit {\_Z}}\right ) \textit {\_Z} \textit {\_f} +c_1 \textit {\_Z} \textit {\_f} +\tan \left (\frac {1}{\textit {\_Z}}\right )+c_1 +\textit {\_Z} -\textit {\_f} \right )}{\textit {\_f}^{2}+1}d \textit {\_f} +c_2 \right ) x \\ \end{align*}

✗ Mathematica

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

NotInvertible : zero divisor