Internal problem ID [2790]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=\frac {4 \,{\mathrm e}^{-2 x}}{x^{2}+1}+2 x^{2}-1} \]
✓ Solution by Maple
Time used: 0.0 (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: 0.58 (sec). Leaf size: 59
DSolve[y''[x]+4*y'[x]+4*y[x]==4*Exp[-2*x]/(1+x^2)+2*x^2-1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{-2 x} \left (8 x \arctan (x)+e^{2 x} x^2-4 \log \left (x^2+1\right )-2 e^{2 x} x+e^{2 x}+2 c_2 x+2 c_1\right ) \]