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Differential Equation w/ Boundary Condition y(1) = e

Hi! I'm brand new to the forum and found that it could be of some help in my engineering studies. I'm currently in Calculus II studying the Diff. Eq. chapter. I am working on a take-home test from my instructor and the problem I'm needing help with (well, I'd really just like it peer reviewed), is a Diff. Eq. with a Boundary Condition. It's requiring me to find the Particular Solution for the equation. It would be of utmost appreciation if anyone could take some time to peer review it, basically just check my work. Also: I used to Linear Method to solve it (e.g. y' + P(x)y = Q(x))

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Thank you very much! :) I think my writing is decently legible.