Question

find a particular solution of y"-y'-6y =e^-x

Answer 1

$y^{''}-y'-6y=e^{-x}.$ (1)

Find a particular solution of (1).

Particular integral is of the form $y=Ce^{-x}.$

This solution must satisfy the differential equation (1):

$(Ce^{-x})''-(Ce^{-x})'-6Ce^{-x}=e^{-x}$

$Ce^{-x}+Ce^{-x}-6Ce^{-x}=e^{-x}$

$2C-6C=1, C=-1/4.$

Then the particular solution of (1) is

$y=-\frac 1 4 e^{-x}.$