Internal problem ID [2777]
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 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=\frac {2 \,{\mathrm e}^{-3 x}}{x^{2}+1}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
dsolve(diff(y(x),x$2)+6*diff(y(x),x)+9*y(x)=2*exp(-3*x)/(x^2+1),y(x), singsol=all)
\[ y \left (x \right ) = c_{2} {\mathrm e}^{-3 x}+x \,{\mathrm e}^{-3 x} c_{1} +\left (2 x \arctan \left (x \right )-\ln \left (x^{2}+1\right )\right ) {\mathrm e}^{-3 x} \]
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 31
DSolve[y''[x]+6*y'[x]+9*y[x]==2*Exp[-3*x]/(x^2+1),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-3 x} \left (2 x \arctan (x)-\log \left (x^2+1\right )+c_2 x+c_1\right ) \]