A sound engineer models a limiter, a device that stops a signal from growing past a fixed level…

Step 1. Separate. $\dfrac{dy}{1 - y^2} = x\,dx$ gives $\tfrac{1}{2}\ln\left|\dfrac{1 + y}{1 - y}\right| = \dfrac{x^2}{2} + c$.

Step 2. The start $y = 0$. Here $|y| < 1$, so $y = \tanh\left(\dfrac{x^2}{2} + c\right)$. At $x = 0$: $0 = \tanh c$, so $c = 0$. Then $y(2) = \tanh 2 = 0.9640$.

Step 3. The start $y = 1$. The constant function $y = 1$ satisfies the equation, because both sides are $0$. The solution through $(0, 1)$ is therefore $y = 1$ for all $x$, and $y(2) = 1$. The formula with $\tanh$ cannot give this, because $\tanh$ never equals $1$.

Step 4. The start $y = 2$. Here $|y| > 1$, so $y = \coth\left(\dfrac{x^2}{2} + c\right)$. At $x = 0$: $\coth c = 2$, so $\tanh c = \dfrac{1}{2}$ and $c = \tfrac{1}{2}\ln 3 = 0.5493$. Then $y(2) = \coth(2 + 0.5493) = \coth 2.5493 = 1.0123$.