23.4.296 problem 299

Internal problem ID [6598]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 299
Date solved : Friday, October 03, 2025 at 02:09:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.289 (sec). Leaf size: 37

ode:=4*diff(y(x),x)^2-2*(y(x)+3*x*diff(y(x),x))*diff(diff(y(x),x),x)+3*x^2*diff(diff(y(x),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: 0.011 (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),