80.3.8 problem 8

Internal problem ID [21172]
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 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 8
Date solved : Thursday, October 02, 2025 at 07:15:24 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }&={\mathrm e}^{t} \left (x^{2}+1\right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 9

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

\[ x = \tan \left ({\mathrm e}^{t}-1\right ) \]

✓ Mathematica. Time used: 0.297 (sec). Leaf size: 15

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

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

✓ Sympy. Time used: 0.182 (sec). Leaf size: 8

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

\[ x{\left (t \right )} = \tan {\left (e^{t} - 1 \right )} \]