Find the general solution of the differential equations:`(dx)/(dy)+2y=sinx`

Explanation

`(dy)/(dx)+2y=sinx`
Here, ` P=2`, `Q=sinx`
`I.F.=e^(intPdx)=e^(int2dx)=e^(2x)`
and general solution :
`y(I.F.)=intQ*(I.F.)dx+c`
`implies ye^(2x)=intsinx*e^(2x)dx+c`…….`(1)`
Let `I=inte^(2x)sinxdx`
`=e^(2x)*(-cosx)-int2e^(2x)(-cosx)dx`
`implies I=-e^(2x)cosx+2[e^(2x)*sinx-int2e^(2x)cosxdx]`
`implies I=-e^(2x)cosx+2e^(2x)sinx-4I`
`implies 5I=e^(2x)(2sinx-cosx)`
`implies I=(1)/(5)e^(2x)(2sinx-cosx)`
From equation `(1)`,
`ye^(2x)=(1)/(5)*e^(2x)(2sinx-cosx)+c`
`implies y=(1)/(5)(2sinx-cosx)+c*e^(-2x)`