First Order Linear Equations: Am I doing it right?

Just seeing if I understand this correctly.

Here is the problem.

I rewrote the problem to this form:

then I found and got

Then I found the and got

So I wrote the general formula as

At this point I wasn't sure if I was finished. I read a tutorial online that said I take it further to find C but they used an equation that was given at the start.

in which case I would input 0 to get C+1 = 1 so C = 0

But that y(0) = 1 was never given.

So am I finished then? Is the answer just the general formula of or is there more yet to do. Or did I just flat out do it wrong? Can someone help me!