64.12.13 problem 13

Internal problem ID [13517]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 13
Date solved : Tuesday, January 28, 2025 at 05:51:00 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\frac {1}{{\mathrm e}^{x}+1} \end{align*}

✓ 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)
 

\[ y = {\mathrm e}^{-2 x} \left (\left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}+1\right )-\ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{x}+\left (x +c_{2} \right ) {\mathrm e}^{x}-c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.060 (sec). Leaf size: 34

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

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