Linear First Order ODE/y' = x + y

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

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

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

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

where:
 * $P \left({x}\right) = -1$
 * $Q \left({x}\right) = x$

Thus:

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