80.2.11 problem 14

Internal problem ID [21154]
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 2. Theory of first order differential equations. Excercise 2.6 at page 37
Problem number : 14
Date solved : Thursday, October 02, 2025 at 07:11:12 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

ode:=diff(x(t),t) = ln(1+x(t)^2); 
dsolve(ode,x(t), singsol=all);
 

\[ t -\int _{}^{x}\frac {1}{\ln \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 33

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

\begin{align*} x(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\log \left (K[1]^2+1\right )}dK[1]\&\right ][t+c_1]\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.269 (sec). Leaf size: 14

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

\[ \int \limits ^{x{\left (t \right )}} \frac {1}{\log {\left (y^{2} + 1 \right )}}\, dy = C_{1} + t \]