65.5.7 problem 7

Internal problem ID [15690]
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 : 7
Date solved : Thursday, October 02, 2025 at 10:23:12 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 15

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

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

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

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

\begin{align*} y(x)&\to \int _2^x\frac {1}{K[1]^2-1}dK[1]+1 \end{align*}

✓ Sympy. Time used: 0.098 (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(2): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} + \frac {\log {\left (3 \right )}}{2} + 1 \]