y’ + y = e^x,

Given y’ + y = e^x and

This is a first order linear differential equation of the form

Here, P = 1 and Q = e^x

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

We have

∴ I.F = e^x

Hence, the solution of the differential equation is,

Recall

However, when x = 0, we have

∴ c = 0

By substituting the value of c in the equation for y, we get

Thus, the solution of the given initial value problem is