22.2.60 problem 60

Internal problem ID [4503]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 60
Date solved : Tuesday, September 30, 2025 at 07:33:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)-y(x) = 1/(1-exp(2*x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\sqrt {1-{\mathrm e}^{2 x}}}{2}+\frac {\left (-\arcsin \left ({\mathrm e}^{x}\right )+2 c_2 \right ) {\mathrm e}^{-x}}{2}+{\mathrm e}^{x} c_1 \]

✓ Mathematica. Time used: 3.061 (sec). Leaf size: 63

ode=D[y[x],{x,2}]-y[x]==1/Sqrt[1-Exp[2*x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} e^{-x} \left (\arctan \left (\frac {e^x}{\sqrt {1-e^{2 x}}}\right )+e^x \sqrt {1-e^{2 x}}-2 c_1 e^{2 x}-2 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.937 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {\operatorname {asin}{\left (e^{x} \right )}}{2}\right ) e^{- x} + \left (C_{2} + \frac {\int \frac {e^{- x}}{\sqrt {- \left (e^{x} - 1\right ) \left (e^{x} + 1\right )}}\, dx}{2}\right ) e^{x} \]