80.5.59 problem D 11

Internal problem ID [21280]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 5. Second order equations. Excercise 5.9 at page 119
Problem number : D 11
Date solved : Thursday, October 02, 2025 at 07:27:38 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 11

ode:=diff(diff(x(t),t),t)-2*diff(x(t),t)*(x(t)-1) = 0; 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {t}{t +1} \]

✗ Mathematica

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

{}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(x(t), t) - Derivative(x(t), (t, 2))/(2*(x(t) - 1)) cannot be solved by the factorable group method