23.4.190 problem 190

Internal problem ID [6492]
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 : 190
Date solved : Tuesday, September 30, 2025 at 03:01:58 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 3 y y^{\prime \prime }&=36 y^{2}+2 {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 26

ode:=3*y(x)*diff(diff(y(x),x),x) = 36*y(x)^2+2*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= -\frac {\left ({\mathrm e}^{4 x} c_1 -c_2 \right )^{3} {\mathrm e}^{-6 x}}{1728} \\ \end{align*}

✓ Mathematica. Time used: 0.607 (sec). Leaf size: 47

ode=3*y[x]*D[y[x],{x,2}] == 36*y[x]^2 + 2*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_2 \left (e^{4 x}+e^{12 c_1}\right ){}^3}{\left (e^{4 x}\right )^{3/2}}\\ y(x)&\to c_2 \left (e^{4 x}\right )^{3/2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-36*y(x)**2 + 3*y(x)*Derivative(y(x), (x, 2)) - 2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(6)*sqrt((-12*y(x) + Derivative(y(x), (x, 2)))*y(x))/2 + De