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(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

y''+3y'+2y = 1/(1+e^{x})

2. Relevant equations

Use variation of parameters (the chapter is about wronskians)

3. 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. . .

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Thank you!

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# Homework Help: Wronskians equations help

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