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atarr3
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Homework Statement
Solve the problem: 4y'' - y = 8e^(.5t)/(2 + e^(.5t))
Homework Equations
Particular solution of Y = X*integral(inverse of X multiplied by G)
Finding eigenvalues and eigenvectors
The Attempt at a Solution
This might be a little too messy for anyone to make sense of, but
I found the eigenvalues first. 4x^2 - 1 = 0, so eigenvalues are +/- 1/2.
Solution for the homogenous is therefore C_1e^(.5t) + C_2e^(-.5t)
The matrix corresponding to the equation X' = AX is:
[0 1]
[.25 0]
The eigenvector for value 1/2 is [1 ] and for -1/2 is [ 1 ]
[.5] [-.5]
So my matrix for the theorem Y = X*integral(inverse of X multiplied by G)
is:
[e^(.5t) e^(-.5t)]
[.5e^(.5t) -.5e^(-.5t)]
So far so good, I think.
The inverse of this is [.5e^(-.5t) e^(-.5t)]
[.5e^(.5t) -e^(.5t) ]
G is [8e^(.5t)/(2 + e^(.5t))]
[ 0 ]
So plugging that into my integral equation thing, I get the integral of
[ 4/(2 + e^(.5t)) ]
[4e^t/(2 + e^(.5t))]
This is where I am stuck and don't know how to integrate.
I'd appreciate any help or advice you can guys can give! Also, please let me know if this problem can be solved a way different from the one I tried to use. Thanks a lot!