Chapter 6 - Inverse Functions - 6.2 Exponential Functions and Their Derivatives - 6.2 Exercises - Page 419: 56

$y''+2y'+y=0$

Show that the function $y=Ae^{-x}+Bxe^{-x}$ satisfies the differential equation $y''+2y'+y=0$. $y'=-Ae^{-x}-Bxe^{-x}+Be^{-x}$ and $y''=Ae^{-x}-Bxe^{-x}-2Be^{-x}$ Now, $y''+2y'+y$ $=Ae^{-x}-Bxe^{-x}-2Be^{-x}+2(-Ae^{-x}-Bxe^{-x}+Be^{-x})+Ae^{-x}+Bxe^{-x}$ $=0$ Hence, the function $y=Ae^{-x}+Bxe^{-x}$ satisfies the differential equation $y''+2y'+y=0$.