Integrating e^3x / (e^x+e^2x) using Variation of Parameters

[Gogeta007]

Im doing my Differential Equation's homework, and I've come across some really hard problems, when it got really complicated I tuoght i was wrong (you know book's problems tend to workout nicely) . . .anyways I got REALLY stuck on this one if anyone can guide me thank you

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 (m²+3m+2) and find that
y_c=c₁e^-2x + c₂e^-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=u₁y₁ + u₂y₂

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 can't 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 doesn't have an integration table) I don't think the book is THAT demanding as into go look it up in another book.
Im thinking maybe I am doing something wrong. . .


==
Thank you!

 

[Roni1985]

e^3x / (e^x+e^2x) is basically e^2x/(1+e^x), right?

did you try integration by parts ?
or first do u=e^x
and see if it helps