Question: Determine all differentiable functions in the form $y$ = $f(x)$ which have the properties:
$f'(x)$=$(f(x))^3$ and $f(0)=2$
What I have done
I set up the differential equation as such
$$ \frac{dy}{dx} = y^3 $$
$$ \int y^{-3} dy = \int dx$$
$$ {-1\over 2y^2} = x + c $$
$$ {1\over 2y^2} = -x-c $$
At $y = 2 , x = 0 , c = -\frac{1}{8}$
$$ {1\over 2y^2} = {1\over 8} - x$$
$$ 2y^2 = {8\over {1-8x}} $$
$$ y^2 = {4\over {1-8x}} $$
$$ y = ±\sqrt{4\over {1-8x}} $$
Is this correct? Are there any limits of x or y to consider?