The general solution of `x(dy)/(dx)=y-xtan((y)/(x))` is . . . .

Options

A. `x^(2)sin((x)/(y))=c`
B. `x sin((x)/(y))=c`
C. `xsin((y)/(x))=c`
D. `x^(2)sin((y)/(x))=c`

Answer

C

Explanation

We have, `x(dy)/(dx)=y-x tan((y)/(x))`
`implies(dy)/(dx)=(y)/(x)-tan((y)/(x))`
Put `(y)/(x)=vimpliesy=vx`
`implies(dy)/(dx)=v+x(dv)/(dx)`
`thereforev+x(dv)/(dx)=v-tanv`
`impliesx(dv)/(dx)=-tanv`
`impliescotvdv=-(dx)/(x)`
`implieslog(sinv)=-log(x)+logc`
`implieslog(sinv*x)=logc`
`implies sin((y)/(x))*x=c`
`impliesx sin((y)/(x))=c`