Find the general solution of the differential equation `(dy)/(dx) + 3y = cos x`

Explanation

`(dy)/(dx) + 3y = cos x` ...(1)
The given differential equation is the linear differential equation of the form `(dy)/(dx) + Py = Q`, where P = 3, Q = cos x
Therefore, I.F. `= e^(int3dx)`
Thus, the solution of (1) is `ye^(3x) = int e^(3x). cos x dx + c`
Let, `int e^(3x) cos x dx = l`
`rArr I = e^(3x).sin x - int 3e^(3x). sin x dx`
`= e^(3x) sin x - 3 (e^(3x).(-cos x) - int 3e^(3x).(-cos x) dx)`
`= e^(3x)(sin x + 3 cos x) - 9l`
`rArr l = (e^(3x))/(10) (sin x + 3 cos x)`
`:.` Solution of (1) is `ye^(3x) = (e^(3x))/(10)(sin x + 3 cos x) + c`
`rArr y = (1)/(10)(sin x + 3 cos x) + ce^(-3x)`