Solution to Linear First Order ODE with Constant Coefficients/With Initial Condition

Theorem
Consider the linear first order ODE with constant coefficients in the form:
 * $(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$

with initial condition $\tuple {x_0, y_0}$

Then $(1)$ has the particular solution:
 * $\displaystyle y = e^{-a x} \int_{x_0}^x e^{a \xi} \map Q \xi \rd \xi + y_0 e^{a \paren {x - x_0} }$

Proof
From Solution to Linear First Order ODE with Constant Coefficients, the general solution to $(1)$ is:
 * $(2): \quad \displaystyle y = e^{-a x} \int e^{a x} \map Q x \rd x + C e^{-a x}$

Let $y = y_0$ when $x = x_0$.

We have:


 * $(3): \quad y_0 = e^{-a x_0} \int e^{a x_0} \map Q {x_0} \rd x_0 + C e^{-a x_0}$

Thus: