Chapter 17 - Second-Order Differential Equations - 17.2 Nonhomogeneous Linear Equations - 17.2 Exercises - Page 1207: 5

$y=e^{2x}(c_{1}cos(\frac{1}{3}x)+c_{2}sin(\frac{1}{3}x)+\frac{1}{37}e^{2x})+\frac{1}{10}e^{-x}$

$y''-4y'+5y=e^{-x}$ $r^{2}-4r+5=0$ $r=2+i$ $y=Ae^{-x}$ $y'=-Ae^{-x}$ $y''=Ae^{-x}$ $Ae^{-x}-4(-Ae^{-x})+5Ae^{-x}=e^{-x}$ $A=\frac{1}{10}$ $y=e^{2x}(c_{1}cos(\frac{1}{3}x)+c_{2}sin(\frac{1}{3}x)+\frac{1}{37}e^{2x})+\frac{1}{10}e^{-x}$