Internal problem ID [13462]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=\frac {{\mathrm e}^{-2 x}}{x^{2}+1}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)+4*diff(y(x),x)+4*y(x)=exp(-2*x)/(1+x^2),y(x), singsol=all)
\[ y = c_{2} {\mathrm e}^{-2 x}+c_{1} x \,{\mathrm e}^{-2 x}-\frac {{\mathrm e}^{-2 x} \left (-2 x \arctan \left (x \right )+\ln \left (x^{2}+1\right )\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 37
DSolve[y''[x]+4*y'[x]+4*y[x]==Exp[-2*x]/(1+x^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{-2 x} \left (2 x \arctan (x)-\log \left (x^2+1\right )+2 (c_2 x+c_1)\right ) \]