Internal problem ID [6322]
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]]
\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=\frac {1}{1+{\mathrm e}^{-x}}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
dsolve(diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=1/(1+exp(-x)),y(x), singsol=all)
\[ y \left (x \right ) = \left (\ln \left ({\mathrm e}^{x}+1\right ) \left ({\mathrm e}^{x}+1\right )+\left (-{\mathrm e}^{x}-1\right ) \ln \left ({\mathrm e}^{x}\right )+{\mathrm e}^{x} c_{1} +c_{2} -1\right ) {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.086 (sec). Leaf size: 34
DSolve[y''[x]-3*y'[x]+2*y[x]==1/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^x \left (2 \left (e^x+1\right ) \text {arctanh}\left (2 e^x+1\right )+c_2 e^x-1+c_1\right ) \]