1. The problem statement, all variables and given/known data Solve the problem: 4y'' - y = 8e^(.5t)/(2 + e^(.5t)) 2. Relevant equations Particular solution of Y = X*integral(inverse of X multiplied by G) Finding eigenvalues and eigenvectors 3. 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!