So I would just like to double check if my solutions are correct for the following two problems:
Thanks in advance, really appreciate it.
Instead of checking your work (which I appreciate you giving), allow me to explain to you how to check your work. Then in the comments to this answer, we can see your progress. (Teach a man to fish, right?)
This is a two-step process. First, plug it back in. If it works out, grand. If not, then you have a big clue that it's not right.
Second, since these are initial value problems, check to make sure that at that value, your function does, in fact, give the correct value.
If these two things work, then your answer is correct. For such computational answers, that's all that's required.