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## Homework Statement

y''+3y'+2y = 1/(1+e

^{x})

## Homework Equations

Use variation of parameters (the chapter is about wronskians)

## The Attempt at a Solution

first we find the complementary equation using the auxiliary equation (m

^{2}+3m+2) and find that

y

_{c}=c

_{1}e

^{-2x}+ c

_{2}e

^{-x}

therefore y_1=e^-2x and y_2=e^-x

then we use the Wronskian of

| y_1 y_2 |

| y'_1 y'_2 | = -e^-3x - ( -2e^-3x) therefore W= e^-3x

W_1= using f(x) = 1/(1+e^x)

| 0 y_2 |

| f(x) y'_2 | = 1 / (e^x+e^2x) = u'

to find y

_{p}=u

_{1}y

_{1}+ u

_{2}y

_{2}

the problem is finding u, since the u' = W_1/W . . .which would yield

e^3x / (e^x+e^2x)

I would have to integrate that, I cant find anyway to do it, I tried integration by parts and partial fraction decomposition.

I was thinking of looking it up in an integration table. But since its a problem from the book (and the book doesnt have an integration table) I dont think the book is THAT demanding as in to go look it up in another book.

Im thinking maybe im doing something wrong. . .

==

Thank you!