Linear First Order ODE/y' = x + y

Theorem
The linear first order ODE:
 * $(1): \quad \dfrac {\d y} {\d x} = x + y$

has the general solution:
 * $y = C e^x - x - 1$

Proof
Rearranging $(1)$:
 * $(2): \quad \dfrac {\d y} {\d x} - y = x$

$(2)$ is a linear first order ODE in the form:
 * $\dfrac {\d y} {\d x} + \map P x y = \map Q x$

where:
 * $\map P x = -1$
 * $\map Q x = x$

Thus:

Thus from Solution by Integrating Factor, $(1)$ can be rewritten as: