The general solution y(x) of the differential equation

\(\rm {dy\over dx}= {\sqrt{1-y^2}\over \sqrt{1-x^2}}\)

Question

Question

Answer 1

Concept:

Separation of variable:

If the equation is such that the variables can be separated, then

Calculation:

The given differential equation is

\(\rm {dy\over dx}= {\sqrt{1-y^2}\over \sqrt{1-x^2}}\)

\(\rm {dy\over \sqrt{1-y^2}}= {dx\over\sqrt{1-x^2}}\)

Integrating both sides, we get

\(\rm \int{dy\over \sqrt{1-y^2}}= \int{dx\over\sqrt{1-x^2}}\)

⇒ sin^-1 y = sin^-1x + c

⇒ sin-1 y = sin^-1x + sin^-1 c

⇒ sin-1 y - sin-1 x = sin-1 c