Why do We differentiate doubly?

Question

The differential equation of all the ellipses centered at the origin is

---

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ after differentiating w.r.t $x,$

$\Leftrightarrow \frac{2 x}{a^{2}}+\frac{2 y y^{\prime}}{b^{2}}=0 \Leftrightarrow \frac{y y^{\prime}}{b^{2}}=-\frac{x}{a^{2}}$

$\Leftrightarrow \frac{\left(y^{\prime}\right)^{2}}{b^{2}}+\frac{y\left(y^{\prime \prime}\right)}{b^{2}}=-\frac{1}{a^{2}}$

$\Leftrightarrow\left(y^{\prime}\right)^{2}+y\left(y^{\prime \prime}\right)^{2}=-\frac{b^{2}}{a^{2}}$

Answer 1

Dividing by $x^2$,

$$\frac1{a^2}+\frac{y^2}{b^2x^2}=\frac1{x^2}$$ and after differentiation,

$$\frac{2xyy'-2y^2}{b^2x^3}=-\frac2{x^3}.$$

Then

$$xyy'-y^2=-b^2$$ gives

$$xy'^2+xyy''-yy'=0.$$

---

A more systematic way is to differentiate the initial equation twice and express that you get a compatible system in $\dfrac1{a^2},\dfrac1{b^2}$:

$$\begin{vmatrix}x^2&y^2&-1\\2x&2yy'&0\\2&2y'^2+2yy''&0\end{vmatrix}=0.$$

Answer 2

The true reason might be found in the context before or after the equations that you have put in your post. Already after one differentiation we have an ordinary differential equation (ODE) for half-ellipses. But all the solutions have their center at origin. By eliminating $x$ through a second differentiation the resulting differential equation will also cover solutions have their center at other points on the $x$-axis. Thus we get an ODE with a larger family of solutions.