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## Homework Statement

What is the value of a such that the solution of the initial-value problem satisfies lim

_{x->infinity}y(x) = 0?

y''+y'=e^(-x), y(0)=1, y'(0)=a

## Homework Equations

## The Attempt at a Solution

Not sure what to do with the missing y term...

y

_{p}=Ae^(-x), y'

_{p}=-A^(-x), y''

_{p}=A^(-x)

Ae^(-x)-A^(-x) = 0 so

y

_{p}=Axe^(-x), y'

_{p}=-Axe^(-x)+Ae^(-x). y''

_{p}=Axe^(-x)-Ae^(-x)-Ae^(-x)

Axe^(-x)-Ae^(-x)-Ae^(-x)-Axe^(-x)+Ae^(-x)=e^(-x)

-Ae^(-x)=e^(-x)

A=-1

The general solution is C

_{1}e^(0)+C

_{2}e^(-x) (????)

y=te^(-x)+C

_{1}e^(0)+C

_{2}e^(-x)

y'=-te^(-x)+e^(-x)-C

_{2}e^(-x)

y(0)=0+C

_{1}-C

_{2}=1

y'(0)=0+1-C

_{2}=a

C

_{2}=1-a

I know this is wrong, multiple numbers can fit into a. Thanks!