40.10.3 problem 12

Internal problem ID [6725]
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
Date solved : Monday, January 27, 2025 at 02:25:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+3 y&=\frac {1}{1+{\mathrm e}^{-x}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 57

dsolve(diff(y(x),x$2)-4*diff(y(x),x)+3*y(x)=1/(1+exp(-x)),y(x), singsol=all)
 

\[ y = \frac {\ln \left (1+{\mathrm e}^{-x}\right ) {\mathrm e}^{x}}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) {\mathrm e}^{3 x}}{2}+\frac {\left (4 c_1 +2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{3 x}}{4}+\frac {{\mathrm e}^{2 x}}{2}+\frac {\left (4 c_2 -1\right ) {\mathrm e}^{x}}{4} \]

✓ Solution by Mathematica

Time used: 0.328 (sec). Leaf size: 49

DSolve[D[y[x],{x,2}]-4*D[y[x],x]+3*y[x]==1/(1+Exp[-x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{4} e^x \left (-4 \left (e^{2 x}-1\right ) \text {arctanh}\left (2 e^x+1\right )+2 e^x+4 c_2 e^{2 x}-1+4 c_1\right ) \]