79.1.3 problem 1 (iii)

Internal problem ID [18411]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 3. Solutions of first-order equations. Exercises at page 47
Problem number : 1 (iii)
Date solved : Thursday, March 13, 2025 at 11:55:50 AM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 10

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

\[ x = \arctan \left (t \right )-\frac {\pi }{4} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 13

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

\[ x(t)\to \arctan (t)-\frac {\pi }{4} \]

✓ Sympy. Time used: 0.136 (sec). Leaf size: 8

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

\[ x{\left (t \right )} = \operatorname {atan}{\left (t \right )} - \frac {\pi }{4} \]