For the differential equation, find a particular solution satisfying the given condition: `(dy)/(dx)+2ytanx=sinx ; y=0` when `x=pi/3`

Explanation

`(dy)/(dx)+2ytanx=sinx`
It is a linear differential equation.
Here, `P=2tanx`, `Q=sinx`
`:. I.F.=e^(intPdx)=e^(2tanxdx)`
`=e^(2logsecx)`
`=e^(log(sec^(2)x)`
`=sec^(2)x`
and general solution :
`ysec^(2)x=intsinx sec^(2)xdx+c`
`=inttanxsecxdx+c`
`implies y sec^(2)x=secx+c`.....`(1)`
Given, that `y=0` at `x=(pi)/(3)`
`:. 0=sec(pi)/(3)+c`
`implies0+2+c`
`implies c=-2`
`:.` From equation `(1)`,
`ysec^(2)x=secx-2`
`implies y=cosx-2cos^(2)x`