Answer to Question #231865 in Differential Equations for tok

Question

verify whether y(x)= 2e^-x + xe^-x is a solution of y" + 2y' + y=0.

Answer 1

$y=2e^{-x} +xe^{-x}$ find

$y'=-2e^{-x}+e^{-x}-xe^{-x}=-e^{-x}-xe^{-x}\\ y''=e^{-x}-e^{-x}+xe^{-x}=xe^{-x}$

Substitute in equation

$xe^{-x}+2\cdot (-e^{-x}+xe^{-x})+2e^{-x}+xe^{-x}=\\ =xe^{-x}-2e^{-x}-2xe^{-x}+2e^{-x}+xe^{-x}=0$

so $y=2e^{-x} +xe^{-x}$is solution of $y''+2y'+y=0$.