This does not strike me as valid. You were good up to the point where you wrote down your second-order DE. I would go this route: whenever you have a DE of the form

you can multiply through by and integrate at least once immediately. In your case, you get

Now you have a first-order DE. Can you continue from here?

Then I'm stuck in both.... there is probably a very simple way of getting the solution out, but my brain just can't turn the concer~ or I've already made a mistake early in my calculations?

Thanks for all the help!