Solve the following differential equation: `cos^2x(dy)/(dx)+y=tanx`

Explanation

`cos^(2)x(dy)/(dx)+y=tanx`
`implies (dy)/(dx)+ysec^(2)x=tanx sec^(2)x`
Here, `P=sec^(2)x` and `Q=tanxsec^(2)x`
Now, `I.F.=e^(intsec^(2)xdx)=e^(tanx)`
and general solution :
`ye^(tanx)=inttanxsec^(2)xe^(tanx)dx+c`
Let `tanx=t`
`implies sec^(2)xdx=dt`
`=intte^(t)dt+c`
`=te^(t)-int1*e^(t)dt+c`
`=te^(t)-e^(t)+c`
`=e^(t)(t-1)+c`
`=e^(tanx)(tanx-1)+c`
`implies y=tanx-1+c*e^(-tanx)`