## How to deal with loss of significance in the case $f(x) = \sqrt{x+2} -\sqrt{x}$?

I would like to evaluate the expression $$f(x) = \sqrt{x+2} -\sqrt{x}$$ with cases when $$x = 3.2 \times 10^{30}$$ and $$x= 3.2 \times 10^{16}$$. I tried using N[Sqrt[x+2] - Sqrt[x], 100] and

ScientificForm[Sqrt[x+2] - Sqrt[x], 100], both yielding 0 as an output. How can I obtain the desired output?

I also tried the method in Making a calculation with high precision by applying .100` as a suffix but my mathematica 12 doesn’t seem to be recognizing it.

## Solution of the differential equation $x dy -y dx = \sqrt(x^2+ y^2)$?

Solution of the differential equation $$x dy -y dx = \sqrt(x^2+ y^2)$$? How do I approach this problem? I noticed that the LHS is in quotient rule form,but I’m not sure how to implement that. Will someone please help? The answer is $$x dy -y dx = \sqrt(x^2+ y^2)$$