Consider the differential equation $f^{\text{'}\text{'}}(x)+2f^{\text{'}}(x)+f(x)=e^{-x}.$ What is the approximate value of the particular solution at $x=9?,r^2+2r+1=e^{-x}$

Hint$-1$: This is a differential equation with constant coefficients!

Hint$-1$: This is a differential equation with constant coefficients!

Hint$-2$: You can take $e^9{90}^2.$

solution

$(r+1)(r+1)$

$f(9)\frac{1}{200000},f(9)\frac{1}{200},f(9)200,f(9)200000$

$f(9)$ should actually $be$ a complex number! $e^{-x}\frac{e^{-x}+-x{e}^{-x}}{1}.$

$(r+1)(r+1)$

$2\frac{e^{-x}}{1}+x{e}^{-x}$

$2*\frac{1}{0^2}+9*\frac{1}{{90}^2}$