40.10.5 problem 14

Internal problem ID [6727]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 15. Linear equations with constant coefficients (Variation of parameters). Supplemetary problems. Page 98
Problem number : 14
Date solved : Monday, January 27, 2025 at 02:25:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\frac {1}{\left (1+{\mathrm e}^{-x}\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 42

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

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