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Gogeta007
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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
y''+3y'+2y = 1/(1+ex)
Use variation of parameters (the chapter is about wronskians)
first we find the complementary equation using the auxiliary equation (m2+3m+2) and find that
yc=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 yp=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!
Homework Statement
y''+3y'+2y = 1/(1+ex)
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
yc=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 yp=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!