23.4.83 problem 83

Internal problem ID [6385]
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 : 83
Date solved : Tuesday, September 30, 2025 at 02:54:56 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 24

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

\[ y = \frac {2 c_1 +\tanh \left (\frac {\ln \left (x \right )-c_2}{2 c_1}\right )}{c_1} \]

✓ Mathematica. Time used: 60.028 (sec). Leaf size: 46

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

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

✗ Sympy

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

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