Hi,
First let me ask you a question...
How are you getting:
x^2+y^2=25
with initial conditions x=0 and y=3.
How did you end up with "25" on the right side?
EDIT: My 'bad', you had y(4)=3 right, not y(0)=3.
The second part:
Following your progress in solving this...
You started with the equation:
dy/dx=y^2-4
Which is in implicit form.
You then were able to solve for 'y' explicitly ie y=f(x).
Now that you know what y is explicitly, you can find dy/dx explicitly by simple differentiation.
You will then know both 'y' and 'dy/dx' so you go back to the original equation, enter those solutions into it (which gives you an explicit form), then solve for the constant 'c'.
So you'll be working with:
f'(x)=f(x)^2-4
where f(x) is your solution for y and f'(x) is its derivative, and you'll be solving for 'c'. All we are doing here is inserting the solutions into the original equation and that gives us an equation where we can solve for 'c'. This is typical of problems like this.
The results for 'c' will be interesting. Once you simplify you'll see the answer, but if not just give a shout.