Linear First Order ODE/y' - y = e^x/y(0) = 0

Theorem
Consider the linear first order ODE:
 * $(1): \quad \dfrac {\d y} {\d x} - y = e^x$

subject to the initial condition:
 * $\map y 0 = 0$

$(1)$ has the particular solution:
 * $y = x e^x$

Proof
$(1)$ is a linear first order ODE in the form:
 * $\dfrac {\d y} {\d x} + \map P x y = \map Q x$

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

Thus:

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

Substituting the initial condition into $(2)$:

which leads to the particular solution:
 * $y = x e^x$