64.12.14 problem 14

Internal problem ID [13518]
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 : 14
Date solved : Tuesday, January 28, 2025 at 05:51:01 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 45

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

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