1)

The solution of the differential equation $\left\{1+x\sqrt{(x^{2}+y^{2})}\right\}dx + \left\{\sqrt{(x^{2}+y^{2})}-1\right\}ydy = 0 is ?


A) x2+y22+13(x2+y2)3/2=C

B) xy23+12(x2+y2)1/2=C

C) xy22+13(x2+y2)3/2=C

D) None of these

Answer:

Option C

Explanation:

Rearranging the equation, we have

dx - ydy + (x2+y2)(xdx+ydy)=0

dxydy+12(x2+y2)d(x2+y2)=0

On integrating, we get

xy22+12tdt=C

{t=x2+y2}

or xy22+13(x2+y2)3/2=C