Internal problem ID [5382]
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: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=\frac {1}{1+{\mathrm e}^{-x}}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 56
dsolve(diff(y(x),x$2)-4*diff(y(x),x)+3*y(x)=1/(1+exp(-x)),y(x), singsol=all)
\[ y = c_{2} {\mathrm e}^{3 x}+c_{1} {\mathrm e}^{x}-\frac {{\mathrm e}^{3 x} \ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {{\mathrm e}^{3 x} \ln \left ({\mathrm e}^{x}\right )}{2}+\frac {\ln \left (1+{\mathrm e}^{-x}\right ) {\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{x}}{4} \]
✓ Solution by Mathematica
Time used: 0.077 (sec). Leaf size: 49
DSolve[y''[x]-4*y'[x]+3*y[x]==1/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} e^x \left (-4 \left (e^{2 x}-1\right ) \text {arctanh}\left (2 e^x+1\right )+2 e^x+4 c_2 e^{2 x}-1+4 c_1\right ) \]