The solution of the differential equation `(dy)/(dx)="tan"((y)/(x))+(y)/(x)` is

Options

A. `cos((y)/(x))=cx`
B. `sin((y)/(x))=cx`
C. `cos((y)/(x))=cy`
D. `sin((y)/(x))=cy`

Answer

B

Explanation

Given, `(dy)/(dx)=tan((y)/(x))+((y)/(x))` . . . (i)
clearly, the given differential equation is homogenous
On putting, `y=Vximplies(dy)/(dx)=V+x(dV)/(dx)` in Eq. (i), we get
`V+x(dV)/(dx)=tanV+V`
`x(dV)/(dx)=tanV`
`(1)/(tanV)dV=(1)/(x)dx`
On integrating both sides, we get
`int(1)/(tanV)dV=int(1)/(x)dx`
`intcotVdV=logx+logc`
`logsinx=logx=logc`
`logsinV=log(xc)`
`sinv=xc`
`sin((y)/(x))=xc`