I've found the same result as in yours. Is it weird or is that the true result? y= Ax+ B/x + C x ln|x|
For the Laplace question, this should also be straight forward. Just apply the laplace transformation to each term. You can look up the transformations on a table.
Once you get the solution in the transformed space, use the table to get the inverse laplace transform of your solution.
If you know how to do it, it is straightforward, but requires a bit of work.
For your test at least know how to setup the question.
ty''+(1-2t)y'-2y=0
Apply laplace denoted by L
L(ty'' + (1-2t)y' - 2y) = L(0)
L(ty'' + y' - 2ty' - 2y) = 0
L(ty'') + L(y') - 2L(ty') - 2L(y) = 0
If you have time, you look up the laplace transforms of these individual terms on a table (probably provided with the exam) that looks like this: http://en.wikipedia.org/wiki/Laplace...s_and_theorems