23.4.252 problem 252

Internal problem ID [6554]
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 : 252
Date solved : Tuesday, September 30, 2025 at 03:04:23 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 73

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

\begin{align*} y &= 1 \\ y &= 0 \\ \frac {2 \left (-\operatorname {signum}\left (y-1\right )\right )^{{2}/{3}} y^{{1}/{6}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {1}{3}, \frac {-y-1}{y-1}\right )}{3 \operatorname {signum}\left (y-1\right )^{{2}/{3}} \left (1-y\right )^{{1}/{3}} \Gamma \left (\frac {2}{3}\right )}-c_1 x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.57 (sec). Leaf size: 153

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{-((\text {$\#$1}-1) \text {$\#$1})} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},1-\text {$\#$1}\right )}{\sqrt [3]{\text {$\#$1}} c_1}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {-3 \sqrt [3]{-((\text {$\#$1}-1) \text {$\#$1})} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},1-\text {$\#$1}\right )}{\sqrt [3]{\text {$\#$1}} (-c_1)}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{-((\text {$\#$1}-1) \text {$\#$1})} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},1-\text {$\#$1}\right )}{\sqrt [3]{\text {$\#$1}} c_1}\&\right ][x+c_2] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(6)*sqrt((y(x) - 1)*y(x)*Derivative(y(x), (x, 2))/(2*y(x) -