, m is a given real number

Given

This is a first order linear differential equation of the form

Here, P = 3 and Q = e^mx

The integrating factor (I.F) of this differential equation is,

We have

∴ I.F = e^3x

Hence, the solution of the differential equation is,

Case (1): m + 3 = 0 or m = –3

When m + 3 = 0, we have e^{(m + 3)x} = e⁰ = 1

⇒ ye^3x = x + c

⇒ ye^3x × e^–3x = (x + c)e^–3x

∴ y = (x + c)e^–3x

Case (2): m + 3 ≠ 0 or m ≠ –3

When m + 3 ≠ 0, we have

Recall

Thus, the solution of the given differential equation is