How does this differential equation define an oscillation from a to b? The differential equation reads:
$$
\dot{y}^2+(y^2-a^2)(y^2-b^2)=0
$$
where $a,b\in \mathcal{R}$ and $a<b$.
It is claimed that this differential equation defines an oscillation from $a$ to $b$.
So I did the research: differentiate w.r.t. $x$. Then I get:
$$
\ddot{y}= (a^2+b^2)y-2y^3
$$ 
Then I looked up in MathWorld, compared with equation(20), I get $k=\sqrt{2-(a^2+b^2)}$.
So I plot the graph using JacobiCN[x,k] in mathematica, it seems not right. 
For example:
I know we need the initial value to get the behavior, since the initial condition is not clear in the article. I suppose it would be $y(0)=a$, can this initial condition plus that differential equation justify the claim stated in the title?
ps: I have little background on elliptic functions so it would be nice if you provide an answer with reference to help me understand.
I wonder anyone can get the following function:
$$
y=b \mathrm{dn}(x,q) \text{  with  } q=1-a^2/b^2
$$
for this function defines an oscillation from $a$ to $b$. 

 A: In the OP, there is an erroneous factor of $2$ appearing in the linear term on $y$.  To arrive at the coveted result, we begin with the ODE
$$\dot y^2(x)=-(y^2(x)-a^2)(y^2(x)-b^2) \tag 1$$
Differentiating both sides of $(1)$ with respect to $x$ yields
$$2\dot y\ddot y=-2y\dot y(y^2-a^2)-2y\dot y(y^2-b^2) \tag 2$$
Now, dividing both side of $(2)$ by $2\dot y$ we obtain
$$\ddot y=-y(y^2-a^2)-y(y^2-b^2)=-y(2y^2-a^2-b^2)=y(a^2+b^2)-2y^3$$
Can you proceed from here?
A: At $y=a$ and $y=b$ the derivative of $y$ is zero according to the diff eq.
For $a \lt y \lt b$, the second summand in the diff eq is negative while for $y \lt a$ or $b \lt y$ it is positive.
Assuming a real valued solution, $y\lt a$ is not possible because we would not get zero, i.e. the diff eq is not satisfied. Likewise $y \gt b$ is not possible. Two positive values cannot add up to zero. 
But $a \lt y \lt b$ is possible and the derivative can then be either positive or negative (depending whether $y^{-1}(a)$ is smaller or larger than $y^{-1}(b)$). Hence, $y$ is monotonic. 
Once $a$ or $b$ is reached, the only way to satisfy the diff eq is that $y$ remains constant.
Conclude by the definition of an oscillation.
