Solve the equation `(dy)/(dx)+(x y)/((1-x^2))xsqrt(y)`

Answer

`sqrt(y)/((1-x^(2))^(1//4)) = -1/3(1-x^(2))^(3//4)+c`

Explanation

`(dy)/(dx) + (xy)/(1-x^(2))=xsqrt(y)`
Dividing by `sqrt(y)`, we get
`1/sqrt(y)(dy)/(dx) + x/(1-x^(2))sqrt(y)=x`………..(1)
Putting `sqrt(y)=v`, we get `1/(2sqrt(y))(dy)/(dx)=(dv)/(dx)`
The given equation transforms to
`(dv)/(dx)+x/(2(1-x^(2)))=1/2x` .........(2)
I.F. `e^(1/2(int[x//(1-x^(2))]dx)`
`e^(-1/4log(1-x^(2))`
`=1//(1-x^(2))^(1//4)`
Therefore,the solution is
`v//(1-x^(2))^(1//4)=1/2int[x//(1-x^(2))^(1//4)]dx+c`
`=-1/4int[(-2x)//(1-x^(2))^(1//4)]dx+c`
`=-1/4(4//3)(1-x^(2))^(3//4)+c`