80.5.62 problem D 14 (a)

Internal problem ID [21283]
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 14 (a)
Date solved : Friday, October 03, 2025 at 07:49:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x x^{\prime \prime }-{x^{\prime }}^{2}+{\mathrm e}^{t} x^{2}&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 11

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

\[ x = {\mathrm e}^{1-{\mathrm e}^{t}} \]

✓ Mathematica. Time used: 0.309 (sec). Leaf size: 14

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

\begin{align*} x(t)&\to e^{1-e^t} \end{align*}

✗ Sympy

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

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