The solution of the differential equation `{1+xsqrt((x^2+y^2))}dx+{sqrt((x^2+y^2))-1}ydy=0` is equal to

Options

A. `x^(2)+y^(2)/2+1/3(x^(2)+y^(2))^(3//2)=c`
B. `x-y^(3)/3+1/2(x^(2)+y^(2))^(1//2)=c`
C. `x-y^(2)/2+1/3(x^(2)+y^(2))^(3//2)=c`
D. None of these

Answer

C

Explanation

`{1+xsqrt((x^(2)+y^(2)))}dx+{sqrt((x^(2)+y^(2))-1)}ydy=0`
or `dx-ydy+sqrt((x^(2)+y^(2))(xdx+ydy))=0`
or `dx-ydy 1/2sqrt((x^(2)+y^(2))d(x^(2)+y^(2)))=0`
Integrating, we have
`x-y^(2)/2+1/2int(sqrt(t)dt) = c, {t=sqrt((x^(2)+y^(2)))}`
Or `x-y^(2)/2+1/3(x^(2)+y^(2))^(3//2)=c`