89.26.2 problem 2

Internal problem ID [24869]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Miscellaneous Exercises at page 177
Problem number : 2
Date solved : Thursday, October 02, 2025 at 10:48:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\frac {1}{\left (1-{\mathrm e}^{2 x}\right )^{{3}/{2}}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x)-y(x) = 1/(1-exp(2*x))^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} c_2 +{\mathrm e}^{-x} c_1 -\sqrt {1-{\mathrm e}^{2 x}} \]

✓ Mathematica. Time used: 3.675 (sec). Leaf size: 35

ode=D[y[x],{x,2}]-y[x]==  1/(1-Exp[2*x])^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 8.765 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + Derivative(y(x), (x, 2)) - 1/(1 - exp(2*x))**(3/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} - \frac {\int \frac {e^{x}}{\left (- \left (e^{x} - 1\right ) \left (e^{x} + 1\right )\right )^{\frac {3}{2}}}\, dx}{2}\right ) e^{- x} + \left (C_{2} + \frac {\int \frac {e^{- x}}{\left (- \left (e^{x} - 1\right ) \left (e^{x} + 1\right )\right )^{\frac {3}{2}}}\, dx}{2}\right ) e^{x} \]