60.7.205 problem 1830 (book 6.239)

Internal problem ID [11755]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1830 (book 6.239)
Date solved : Thursday, March 13, 2025 at 09:44:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} {y^{\prime \prime }}^{2}-2 \left (3 x y^{\prime }+y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.958 (sec). Leaf size: 37

ode:=3*x^2*diff(diff(y(x),x),x)^2-2*(3*x*diff(y(x),x)+y(x))*diff(diff(y(x),x),x)+4*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x^{1+\frac {2 \sqrt {3}}{3}} c_{1} \\ y &= 0 \\ y &= \frac {c_{1}^{2} x^{2}}{c_{2}}+c_{1} x +c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 29

ode=4*D[y[x],x]^2 - 2*(y[x] + 3*x*D[y[x],x])*D[y[x],{x,2}] + 3*x^2*D[y[x],{x,2}]^2 == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {c_1{}^2 x^2}{c_2}+c_1 x+c_2 \\ y(x)\to \text {Indeterminate} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2))**2 - (6*x*Derivative(y(x), x) + 2*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 -3*x*Derivative(y(x), (x, 2))/4 - sqrt((-3*x**2*Derivative(y(x), (x, 2)) + 8*y(x))*Derivative(y(x), (x, 2)))/4 + Derivative(y(x), x) cannot be solved by the factorable group method