74.3.11 problem 12

Internal problem ID [15792]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 12
Date solved : Thursday, March 13, 2025 at 06:20:18 AM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 6

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

\[ y = \arctan \left (t \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 20

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

\[ y(t)\to \int _0^t\frac {1}{K[1]^2+1}dK[1] \]

✓ Sympy. Time used: 0.156 (sec). Leaf size: 5

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

\[ y{\left (t \right )} = \operatorname {atan}{\left (t \right )} \]