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 : Monday, January 27, 2025 at 09:21:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 37

dsolve(diff(y(x),x$2)-y(x)=1/sinh(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (-x +2 c_{1} -\ln \left (\sinh \left (x \right )\right )\right ) {\mathrm e}^{-x}}{2}-\frac {{\mathrm e}^{x} \left (x -2 c_{2} -\ln \left (\sinh \left (x \right )\right )\right )}{2} \]

✓ Solution by Mathematica

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

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

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