89.24.14 problem 14

Internal problem ID [24851]
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. Exercises at page 171
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:48:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 70

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

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

✓ Mathematica. Time used: 0.13 (sec). Leaf size: 77

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

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

✓ Sympy. Time used: 0.701 (sec). Leaf size: 42

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

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