28.2.55 problem 55

Internal problem ID [4498]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 55
Date solved : Sunday, March 30, 2025 at 03:24:06 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\frac {1}{\sinh \left (x \right )} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)-y(x) = 1/sinh(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x} c_2 +{\mathrm e}^{x} c_1 -x \cosh \left (x \right )+\ln \left (\sinh \left (x \right )\right ) \sinh \left (x \right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 53

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

\[ y(x)\to e^x \text {arctanh}\left (1-2 e^{2 x}\right )-\frac {1}{2} e^{-x} \log \left (1-e^{2 x}\right )+c_1 e^x+c_2 e^{-x} \]

✓ Sympy. Time used: 0.806 (sec). Leaf size: 34

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

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