80.5.61 problem D 13

Internal problem ID [21282]
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 13
Date solved : Thursday, October 02, 2025 at 07:27:39 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x x^{\prime \prime }-2 {x^{\prime }}^{2}-x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 21

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

\begin{align*} x &= 0 \\ x &= \frac {\sec \left (t \right )}{c_1 \tan \left (t \right )-c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.165 (sec). Leaf size: 13

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

\begin{align*} x(t)&\to c_2 \sec (t+c_1) \end{align*}

✗ Sympy

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

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