28.2.65 problem 65

Internal problem ID [4508]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 65
Date solved : Monday, January 27, 2025 at 09:22:07 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 30

dsolve(diff(y(x),x$2)+diff(y(x),x)=1/(exp(x)+1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.085 (sec). Leaf size: 33

DSolve[D[y[x],{x,2}]+D[y[x],x]==1/(Exp[x]+1),y[x],x,IncludeSingularSolutions -> True]
 

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