- #1

- 136

- 0

[tex]\frac{dx}{\sqrt{1-x^2}} + \frac{dy}{\sqrt{1-y^2}} = 0[/tex]

It can be rewritten as:

[tex]\sqrt{1-y^2}dx + \sqrt{1-x^2}dy = 0[/tex]

One solution of this equation (besides arcsin(x) + arcsin(y) = C), is given by:

[tex]x\sqrt{1-y^2} + y\sqrt{1-x^2} = C[/tex]

However, I'm wondering how this equation can be derived from the equation immediately preceding it (i.e., how the "solution", equation 3 above, can be derived from equation 2)?

It looks like it was simply integrated, term-by-term, that is the "dx" term was integrated to give "x" and the "dy" term was integrated to give "y", but usually this can only be done if the D.E. is exact, and this equation does not appear to be exact.

So how is one justified in arriving at that answer, if the equation is not exact?