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 homogeneous
On putting, `y = Vx`
`rArr (dy)/(dx) = V +x'(dV)/(dx)` in Eq. (i), we get
`V + x'(dV)/(dx) = tan V +V`
`rArr x'(dV)/(dx) = tanV`
`rArr (1)/(tanV) dV = 1/x dx`
On integrating both sides, we get
`int(1)/(tanV) dV = int 1/x dx`
`rArr int cot V dV = log x +log c`
`rArr log sin x = log x +log c`
`rArr log sin V = log (x C)`
`rArr sin v = xc`
`:. sin(y/x) = xc`