23.4.163 problem 163

Internal problem ID [6465]
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 : 163
Date solved : Tuesday, September 30, 2025 at 03:01:32 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 {y^{\prime }}^{2}+\left (1-y\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 22

ode:=2*diff(y(x),x)^2+(1-y(x))*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 1 \\ y &= \frac {c_1 x +c_2 -1}{c_1 x +c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.115 (sec). Leaf size: 37

ode=2*D[y[x],x]^2 + (1 - y[x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 x-1+c_2 c_1}{c_1 (x+c_2)}\\ y(x)&\to 1\\ y(x)&\to \text {Indeterminate} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 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(2)*sqrt((y(x) - 1)*Derivative(y(x), (x, 2)))/2 + Derivativ