Answer
$y''+2y'+y=0$
Work Step by Step
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 $y''+2y'+y=0$.