23.4.187 problem 187

Internal problem ID [6489]
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 : 187
Date solved : Tuesday, September 30, 2025 at 03:01:54 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 y y^{\prime \prime }&=y^{2} \left (1-3 y^{2}\right )+6 {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 41

ode:=2*y(x)*diff(diff(y(x),x),x) = y(x)^2*(1-3*y(x)^2)+6*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {1}{\sqrt {-2 c_2 \cos \left (x \right )+2 c_1 \sin \left (x \right )+3}} \\ y &= -\frac {1}{\sqrt {-2 c_2 \cos \left (x \right )+2 c_1 \sin \left (x \right )+3}} \\ \end{align*}

✓ Mathematica. Time used: 60.371 (sec). Leaf size: 247

ode=2*y[x]*D[y[x],{x,2}] == y[x]^2*(1 - 3*y[x]^2) + 6*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {-\frac {3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}+3}{-9+4 c_1 \tan ^2(x+c_2)}}\\ y(x)&\to \sqrt {-\frac {3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}+3}{-9+4 c_1 \tan ^2(x+c_2)}}\\ y(x)&\to -\sqrt {\frac {-3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}-3}{-9+4 c_1 \tan ^2(x+c_2)}}\\ y(x)&\to \sqrt {\frac {-3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}-3}{-9+4 c_1 \tan ^2(x+c_2)}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(1 - 3*y(x)**2)*y(x)**2 + 2*y(x)*Derivative(y(x), (x, 2)) - 6*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(6)*sqrt((3*y(x)**3 - y(x) + 2*Derivative(y(x), (x, 2)))*y(