Question
The solution of the equation $$\frac{{dy}}{{dx}} = \sqrt {\frac{{1 - {y^2}}}{{1 - {x^2}}}} $$ is :
A.
$${\sin ^{ - 1}}y - {\sin ^{ - 1}}x = c$$
B.
$${\sin ^{ - 1}}y\,{\sin ^{ - 1}}x = c$$
C.
$${\sin ^{ - 1}}\left( {xy} \right) = 2$$
D.
none of these
Answer :
$${\sin ^{ - 1}}y - {\sin ^{ - 1}}x = c$$
Solution :
$$\eqalign{
& \frac{{dy}}{{dx}} = \sqrt {\frac{{1 - {y^2}}}{{1 - {x^2}}}} \cr
& \therefore \,\frac{{dy}}{{\sqrt {1 - {y^2}} }} = \frac{{dx}}{{\sqrt {1 - {x^2}} }} \cr
& \Rightarrow \int {\frac{{dy}}{{\sqrt {1 - {y^2}} }}} = \int {\frac{{dx}}{{\sqrt {1 - {x^2}} }}} \cr
& \Rightarrow {\sin ^{ - 1}}y = {\sin ^{ - 1}}x + c \cr
& \therefore \,{\sin ^{ - 1}}y - {\sin ^{ - 1}}x = c \cr} $$