23.4.193 problem 193

Internal problem ID [6495]
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 : 193
Date solved : Tuesday, September 30, 2025 at 03:02:04 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 4 y y^{\prime \prime }&=12 y^{2}+3 {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 31

ode:=4*y(x)*diff(diff(y(x),x),x) = 12*y(x)^2+3*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\left ({\mathrm e}^{\sqrt {3}\, x} c_1 -c_2 \right )^{4} {\mathrm e}^{-2 \sqrt {3}\, x}}{2304} \\ \end{align*}

✓ Mathematica. Time used: 0.524 (sec). Leaf size: 58

ode=4*y[x]*D[y[x],{x,2}] == 12*y[x]^2 + 3*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-12*y(x)**2 + 4*y(x)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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