Find the general solution of the differential equation

Before starting solving the differential equation, we should know the type of differential equation.

This is the linear type differential equation,

– y = sinx

It is of the form + Py = Q where P and Q are either constants or function of x.

To solve these type of equations, we need to first find the integrating factor which is = and multiply both sides of the general form of equa the tion and after that integrate the whole equation.

First, convert the given equation in general form equation,

– y = sinx

P = - 1 and Q = sinx

Now find the I.F. = = e ^{- x}

Multiplying both sides by I.F. we get,

e ^{- x} – e ^{- x} y = e ^{- x} sinx

Integrating both sides with respect to x, we get,

e ^{- x} y + e ^{- x} y = (–1) e ^{- x} (cosx + sinx) + C

2 e ^{- x} y = (–1) e ^{- x} (cosx + sinx) + C