73.16.6 problem 24.1 (f)

Internal problem ID [15518]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 24. Variation of parameters. Additional exercises page 444
Problem number : 24.1 (f)
Date solved : Tuesday, January 28, 2025 at 08:00:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=\frac {{\mathrm e}^{-2 x}}{x^{2}+1} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 50

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

\[ y(x)\to \frac {1}{2} e^{-2 x} \left (2 x \int _1^x\frac {1}{K[1]^2+1}dK[1]-\log \left (x^2+1\right )+2 (c_2 x+c_1)\right ) \]