With regard to your solution of 1): a. You forgot the constant of integration on the RHS. b. I think your constants might be a little bit off. Check them again, especially with respect to your integration by parts.
With regard to your solution of 2): I think your constants are off again. Check the same thing as with your solution of #1.
In general, you've got the right method of solution. Just make sure the details are correct!