Internal problem ID [5384]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-y=\frac {1}{\left (1+{\mathrm e}^{-x}\right )^{2}}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)-y(x)=1/(1+exp(-x))^2,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{1} +\frac {{\mathrm e}^{x}}{2}-1+{\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}+1\right )+\frac {{\mathrm e}^{-x}}{2} \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 42
DSolve[y''[x]-y[x]==1/(1+Exp[-x])^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{-x} \left (-2 e^x+2 \log \left (e^x+1\right )+2 c_1 e^{2 x}+1+2 c_2\right ) \]