Find the general solution of the differential equation
`(dy)/(dx) =sin^(-1)x.`

Explanation

We have ,
`(dy)/(dx) = sin^(-1)x`
`rArr dy=sin^(-1)xdx" " ` [separating the variables]
`rArr int dy=int sin^(-1)x dx " " `[integrating both sides]
`rArr y=int {(underset(I)(sin^(-1)x))*underset(II)(1)}dx + C, ` where C is an arbitrary constant
`=(sin^(-1)x)*x-int(1)/(sqrt(1-x^(2)))*x dx + C " " `[integrating by parts]
`=(sin^(-1)x)*x + (1)/(2) int (-2x)/(sqrt(1-x^(2)))dx + C`
`=(sin^(-1)x) *x+(1)/(2) int (1)/(sqrt(t))dt + C, " where "(1-x^(2))=t`
`=(sin^(-1)x)*x + (1)/(2) xx 2 sqrt(t) + C`
`=(sin^(-1)x)x +sqrt(1-x^(2))+C.`
Hence, `y=(sin^(-1)x)x+sqrt(1-x^(2))+C` is the required solution.
Hence, `y=x(sin^(-1)x)+sqrt(1-x^(2))+C` is the required solution of the given differential equation.