68.4.51 problem 51

Internal problem ID [17231]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 51
Date solved : Thursday, October 02, 2025 at 01:59:14 PM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 8

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

\[ y = \arctan \left (x \right )+1 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 22

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

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

✓ Sympy. Time used: 0.089 (sec). Leaf size: 7

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

\[ y{\left (x \right )} = \operatorname {atan}{\left (x \right )} + 1 \]