23.4.249 problem 249

Internal problem ID [6551]
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 : 249
Date solved : Tuesday, September 30, 2025 at 03:04:20 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 23

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

\begin{align*} y &= 1 \\ y &= 0 \\ y &= \tanh \left (\frac {c_1 x}{2}+\frac {c_2}{2}\right )^{2} \\ \end{align*}

✓ Mathematica. Time used: 3.02 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to \tanh ^2\left (\frac {1}{2} c_1 (x+c_2)\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2)*sqrt((y(x) - 1)*y(x)*Derivative(y(x), (x, 2))/(3*y(x) -