The solution of the differential equation `x^(4)(dy)/(dx)+x^(3)y+"cosec"(xy)=0` is equal to

Options

A. `2cos (xy)+x^(-2)=C`
B. ` 2 cos (xy)+y^(-2)=C`
C. ` 2 sin (xy) +y^(-2) = C`
D. ` 2 sin (xy) +y^(-2) = C`

Answer

a

Explanation

` x^(4) (dy)/(dx) +x^(3)y+"cosec"(xy)=0`
` rArr x^(4)dy +x^(3)y + "cosec" (xy)=0`
` rArr x^(3)( x dy = y dx) + "cosec" dx = 0`
` rArr x^(3)d(xy)+"cosec " (xy)dx =0`
` rArr (d(xy))/("cosec"(xy)) +(dx)/(x^(3))=0`
On integrating both sides , we get
` int sin (xy) d(xy) + int x^(-3)dx = 0 `
` rArr - cos (xy)+ ((x^(-2))/-2)=C_(1)`
` :. 2 cos (xy)+x^(-2)=C " " [ :' C = -2C_(1)]`