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This is from Tenenbaum & Pollard's "Ordinary Differential Equations" book, chapter 1, Exercise 2, problem # 15:

I don't know why we can't apply the method of implicit differentiation to the relation:

$$\sqrt{x^2 -y^2} + \arccos{(\frac{x}{y})} = 0 \quad (1)$$

For this relation there is as function $y=g(x) = x$ isn't?

but for relation:

$$\sqrt{x^2 -y^2} + \arcsin{(\frac{x}{y})} = 0 \quad (2)$$

but for second I know we can't because for $y = x$: $$\arcsin(\frac{x}{x}) = \frac{\pi}{2}$$ and for y = -x $$\arcsin(\frac{x}{-x}) = -\frac{\pi}{2}$$ and equation (2) isn't true. But I why we can't apply method of implicit differentiation for the (1) relation?

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    Some context might be helpful. If you want to find the derivative of $y$ respect to $x$, implicit differentiation should be fine for both. And where does $y=x$ or $y=-x$ come from? – 2017-01-04
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    From here: http://math.stackexchange.com/questions/1589942/how-to-find-an-implicit-function-from-this-relation – 2017-01-04
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    Interesting. You can still apply implicit differentiation. At some point though, the domain problem will appear again. For the one with $\arcsin$, the function does not exist, because neither of $y=x$ or $y=-x$ satisfy the equation. – 2017-01-04

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