20.9.17 problem Problem 17

Internal problem ID [3761]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 17
Date solved : Monday, January 27, 2025 at 08:00:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.706 (sec). Leaf size: 59

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

\[ y(x)\to \frac {1}{2} e^{-2 x} \left (8 x \arctan (x)+e^{2 x} x^2-4 \log \left (x^2+1\right )-2 e^{2 x} x+e^{2 x}+2 c_2 x+2 c_1\right ) \]