60.7.195 problem 1818 (book 6.227)

Internal problem ID [11745]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1818 (book 6.227)
Date solved : Sunday, March 30, 2025 at 09:08:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x y^{\prime }-y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 44

ode:=(-y(x)+x*diff(y(x),x))*diff(diff(y(x),x),x)+4*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int _{}^{\ln \left (x \right )}\left ({\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-1\right ) {\mathrm e}^{\textit {\_Z}}+c_1 \,{\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-\textit {\_b} \,{\mathrm e}^{\textit {\_Z}}+2\right )}-1\right )d \textit {\_b} +c_2} \\ \end{align*}

✓ Mathematica. Time used: 60.523 (sec). Leaf size: 41

ode=4*D[y[x],x]^2 + (-y[x] + x*D[y[x],x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} c_2 e^{-2-W\left (\frac {2 x}{e^2 c_1}\right )} \left (2+W\left (\frac {2 x}{e^2 c_1}\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x*Derivative(y(x), x) - y(x))*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x*Derivative(y(x), (x, 2))/8 - sqrt((x**2*Derivative(y(x), (x, 2)) + 16*y(x))*Derivative(y(x), (x, 2)))/8 + Derivative(y(x), x) cannot be solved by the factorable group method