23.6.8 problem 8

Internal problem ID [6807]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 6. THE EQUATION IS NONLINEAR AND OF ORDER GREATER THAN TWO, page 427
Problem number : 8
Date solved : Friday, October 03, 2025 at 02:09:53 AM
CAS classification : [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\begin{align*} 3 y^{2}+18 x y y^{\prime }+9 x^{2} {y^{\prime }}^{2}+9 x^{2} y y^{\prime \prime }+3 x^{3} y^{\prime } y^{\prime \prime }+x^{3} y y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 56

ode:=3*y(x)^2+18*x*y(x)*diff(y(x),x)+9*x^2*diff(y(x),x)^2+9*x^2*y(x)*diff(diff(y(x),x),x)+3*x^3*diff(y(x),x)*diff(diff(y(x),x),x)+x^3*y(x)*diff(diff(diff(y(x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\sqrt {-x \left (c_1 \,x^{2}+2 c_3 x -2 c_2 \right )}}{x^{2}} \\ y &= -\frac {\sqrt {-x \left (c_1 \,x^{2}+2 c_3 x -2 c_2 \right )}}{x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.133 (sec). Leaf size: 60

ode=3*y[x]^2 + 18*x*y[x]*D[y[x],x] + 9*x^2*D[y[x],x]^2 + 9*x^2*y[x]*D[y[x],{x,2}] + 3*x^3*D[y[x],x]*D[y[x],{x,2}] + x^3*y[x]*D[y[x],{x,3}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}}\\ y(x)&\to \frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*y(x)*Derivative(y(x), (x, 3)) + 3*x**3*Derivative(y(x), x)*Derivative(y(x), (x, 2)) + 9*x**2*y(x)*Derivative(y(x), (x, 2)) + 9*x**2*Derivative(y(x), x)**2 + 18*x*y(x)*Derivative(y(x), x) + 3*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*Derivative(y(x), (x, 2)) + sqrt(x**