Find the general solution of the differential equation
`x sqrt(1-y^(2))dx + y sqrt(1-x^(2)) dy =0.`

Explanation

We have
`x sqrt(1-y^(2))dx + y sqrt(1-x^(2)) dy =0`
`rArr (x)/(sqrt(1-x^(2)))dx+(y)/(sqrt(1-y^(2)))dy=0 " " `[on separating the variables]
`rArr int (x)/(sqrt(1-x^(2)))dx+int (y)/(sqrt(1-y^(2)))dy=C " " ` [integrating both sides]
`rArr -(1)/(2) *int ((-2x))/(sqrt(1-x^(2)))dx -(1)/(2)*int ((-2y))/(sqrt(1-y^(2)))dy=C`
`rArr -(1)/(2)*int (1)/(sqrt(t)) dt - (1)/(2) *int (1)/(sqrt(s))ds =C, " where " (1-x^(2))=t and (1-y^(2)) =s`
`rArr -(1)/(2) int t^(-1//2) dt -(1)/(2)int s^(-1//2) ds =C`
`rArr -sqrt(t) -sqrt(s) =C`
`rArr sqrt(t)+sqrt(s)=k" " ["where "k= -C]`
`rArr sqrt(1-x^(2)) + sqrt(1-y^(2))=k. `
Hence, `sqrt(1-x^(2))+sqrt(1-y^(2))=k ` is the required solution.