68.3.11 problem 12

Internal problem ID [17158]
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, October 02, 2025 at 01:45:28 PM
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: 0.038 (sec). Leaf size: 6

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

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

✓ Mathematica. Time used: 0.003 (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]
 

\begin{align*} y(t)&\to \int _0^t\frac {1}{K[1]^2+1}dK[1] \end{align*}

✓ Sympy. Time used: 0.088 (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)
 

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