71.5.8 problem 8

Internal problem ID [14325]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.3.1, page 57
Problem number : 8
Date solved : Wednesday, March 05, 2025 at 10:45:48 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: 0.010 (sec). Leaf size: 10

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 22

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

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

✓ Sympy. Time used: 0.166 (sec). Leaf size: 22

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)
 

\[ y{\left (x \right )} = \frac {\log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} + 1 - \frac {i \pi }{2} \]