28.2.60 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 : Monday, January 27, 2025 at 09:21:54 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 36

dsolve(diff(y(x),x$2)-y(x)=1/sqrt(1-exp(2*x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 4.999 (sec). Leaf size: 63

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

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