What am I doing wrong? Integration and limits I need some help identifying what I'm doing wrong here..
What is the limit of $y(x)$ when $x→∞$ if $y$ is given by:
$$y(x) = 10 + \int_0^x \frac{22(y(t))^2}{1 + t^2}\,dt$$
What i've done:
1) Integrating on both sides(and using the Fundamental Theorem of Calculus):
$$\frac{dy}{dx} = 0 + \frac{22(y(x))^2}{1 + x^2}$$
2)
$$\frac{-1}{22y} = \arctan x$$
And after moving around and stuff I end up with the answer:  $\quad\dfrac{-1}{11 \pi}.$
What's wrong?

To add my comment above, the answer should be $\frac{-1}{22y} + \frac{1}{22\cdot 10} = \text{atan}(x)$. This reduces to $y(x) = \frac{-1}{22}\cdot (\text{atan}(x) - \frac{1}{220})^{-1}$. On a rigorous level, we know that this holds since $y(x)\geq 10\; \forall\; x$. Now $\text{lim}\; f(x) = \frac{-1}{22}\cdot (\frac{\pi}{2} - \frac{1}{220})^{-1}$.
