Internal problem ID [5569]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF
PARAMETERS. Page 71
Problem number: 1(f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-3 y^{\prime }+2 y-\frac {1}{1+{\mathrm e}^{-x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=1/(1+exp(-x)),y(x), singsol=all)
\[ y \relax (x ) = \left (c_{1} {\mathrm e}^{x}-\ln \left ({\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{x}+\ln \left ({\mathrm e}^{x}+1\right ) \left ({\mathrm e}^{x}+1\right )-1+c_{2}\right ) {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 34
DSolve[y''[x]-3*y'[x]+2*y[x]==1/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^x \left (2 \left (e^x+1\right ) \tanh ^{-1}\left (2 e^x+1\right )+c_2 e^x-1+c_1\right ) \\ \end{align*}