Internal problem ID [2281]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y^{\prime }+4 y-\frac {4 \,{\mathrm e}^{-2 x}}{x^{2}+1}-2 x^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 46
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)
\[ y \relax (x ) = c_{2} {\mathrm e}^{-2 x}+{\mathrm e}^{-2 x} x c_{1}-2 \,{\mathrm e}^{-2 x} \ln \left (x^{2}+1\right )+4 \,{\mathrm e}^{-2 x} \arctan \relax (x ) x +\frac {\left (x -1\right )^{2}}{2} \]
✓ Solution by Mathematica
Time used: 0.218 (sec). Leaf size: 41
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]
\begin{align*} y(x)\to \frac {1}{2} (x-1)^2+e^{-2 x} \left (4 x \text {ArcTan}(x)-2 \log \left (x^2+1\right )+c_2 x+c_1\right ) \\ \end{align*}