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BiGyElLoWhAt

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## Homework Statement

Find the green's function for y'' +4y' +3y = 0 with y(0)=y'(0)=0 and use it to solve y'' +4y' +3' =e^-2x

## Homework Equations

##y = \int_a^b G*f(z)dz##

## The Attempt at a Solution

##\lambda^2 + 4\lambda + 3 = 0 \to \lambda = -1,-3##

##G(x,z) = \left\{ \begin{array}{ll}

Ae^{-x} + Be^{-3x} & z<x \\

Ce^{-x} + De^{-3x} & x<z

\end{array} \right. ##

y(0) = 0 -> A=-B

y'(0) =0 -> A=B=0

continuity:

##Ce^{-z} +De^{-3z}=0 \to C = -De^{-2z}##

##-Ce^{-z} -3De^{-3z} = 1 \to [De^{-2^z}]e^{-z} - 3De^{-3z} = -2De^{-3z} = 1##

##D=-1/2e^{3z}## & ##C= 1/2e^z##

##G(x,z) = \left\{ \begin{array}{ll}

0 & z<x \\

1/2e^ze^{-x} - 1/2e^{3z}e^{-3x} & x<z

\end{array} \right. ##

The problem comes in when I go to integrate to get the solution.

##y = \int_x^{\infty} [ 1/2e^{-(x+z)} - 1/2 e^{z-3x}]dz##

The second integral is divergent in z.

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