80.1.22 problem 19

Internal problem ID [21140]
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 1. First order linear differential equations. Excercise 1.5 at page 13
Problem number : 19
Date solved : Thursday, October 02, 2025 at 07:09:48 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }&=\frac {x}{t^{2}+1} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 7

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

\[ x = {\mathrm e}^{\arctan \left (t \right )} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 9

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

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

✓ Sympy. Time used: 0.122 (sec). Leaf size: 7

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

\[ x{\left (t \right )} = e^{\operatorname {atan}{\left (t \right )}} \]