# Another exasperating calculation of $\frac{d^2y}{dx^2}$

Suppose $ax^2+2hxy+by^2=1$.I have to find $\frac{d^2y}{dx^2}$.

I have found the answer using the method described below.

Direct differentiation yields $\frac{dy}{dx}=-\frac{ax+hy}{hx+by}\dots (1)$.

We have $ax+hy+(hx+by)\frac{dy}{dx}=0$. Differentiating yet again, $a+2h\frac{dy}{dx}+b(\frac{dy}{dx})^2+\frac{d^2y}{dx^2}(hx+by)=0$. Substituting the value of $\frac{dy}{dx}$ obtained above in $(1)$, after a lot of calculation, $$\frac{d^2y}{dx^2}=\frac{h^2-ab}{(hx+by)^3}$$ I am seeking a quicker method; this one takes a lot of time and is not appropriate for my exams.So does anyone have any shorter method?

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I don't think there's a method that's fundamentally more direct, but you could organize the calculations a bit more efficiently by writing the equation as

$$r^\top Ar=1$$

with

$$r=\pmatrix{x\\y}$$

and

$$A=\pmatrix{a&h\\h&b}\;.$$

Then

$$r^\top Ar'=0$$

and

$$r^\top Ar''+r'^\top Ar'=0$$

with

$$r'=\pmatrix{1\\y'}$$

and

$$r''=\pmatrix{0\\y''}\;.$$

That simplifies the differentiations, and it's only solving for $y'$ and $y''$ that you now have to do explicitly; I don't see a way around that.

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