23.4.229 problem 229

Internal problem ID [6531]
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 : 229
Date solved : Tuesday, September 30, 2025 at 03:02:44 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 3 x y^{2}+6 x^{2} y y^{\prime }+x^{3} {y^{\prime }}^{2}+x^{3} y y^{\prime \prime }&=a \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 50

ode:=3*x*y(x)^2+6*x^2*y(x)*diff(y(x),x)+x^3*diff(y(x),x)^2+x^3*y(x)*diff(diff(y(x),x),x) = a; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.272 (sec). Leaf size: 58

ode=3*x*y[x]^2 + 6*x^2*y[x]*D[y[x],x] + x^3*D[y[x],x]^2 + x^3*y[x]*D[y[x],{x,2}] == a; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a + x**3*y(x)*Derivative(y(x), (x, 2)) + x**3*Derivative(y(x), x)**2 + 6*x**2*y(x)*Derivative(y(x), x) + 3*x*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-3*x**2*y(x) + sqrt(x**3*(a - x**3*y(x)*D