Chapter 17 - Second-Order Differential Equations - 17.2 Nonhomogeneous Linear Equations - 17.2 Exercises - Page 1207: 6

$y=e^{-x}(c_{1}cos(2x)+c_{2}sin(2x))+\frac{1}{5}+\frac{1}{8}e^{x}$

$y''+2y'+5y=1+e^x$ $r^2+2r+5=0$ $r=-1+2i$ $y=e^{-x}(c_{1}cos(2x)+c_{2}sin(2x))$ $y=A+Be^x$ $y'=Be^x$ $y''=Be^x$ $Be^x+2Be^x+5A+5Be^x=1+e^x$ $A=\frac{1}{5}$ $B=\frac{1}{8}$ $y=e^{-x}(c_{1}cos(2x)+c_{2}sin(2x))+\frac{1}{5}+\frac{1}{8}e^{x}$