60.7.157 problem 1774 (book 6.183)

Internal problem ID [11707]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1774 (book 6.183)
Date solved : Thursday, March 13, 2025 at 09:23:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.091 (sec). Leaf size: 30

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

\[ y = \frac {x \left (4 c_{2}^{2} \ln \left (x \right )^{2}+4 \ln \left (x \right ) c_{1} c_{2} +c_{1}^{2}+1\right )}{4 c_{2}} \]

✓ Mathematica. Time used: 0.549 (sec). Leaf size: 49

ode=y[x]^2 - x^2*(1 + D[y[x],x]^2) + 2*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 {x \left (c_1{}^2 \log ^2(x)-2 c_2 c_1{}^2 \log (x)+4+c_2{}^2 c_1{}^2\right )}{4 c_1} \\ y(x)\to \text {Indeterminate} \\ \end{align*}

✗ Sympy

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

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