Solution to Linear First Order ODE with Constant Coefficients/Proof 2

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Theorem

A linear first order ODE with constant coefficients in the form:

$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$

has the general solution:

$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$


Proof

This is a specific instance of Solution to Linear First Order Ordinary Differential Equation:

A linear first order ordinary differential equation in the form:

$\dfrac {\d y} {\d x} + \map P x y = \map Q x$

has the general solution:

$\ds y = e^{-\int P \rd x} \paren {\int Q e^{\int P \rd x} \rd x + C}$


In this instance, we have:

$\map P x = a$

Hence:

$\ds \int \map P x = a x$

and the result follows.

$\blacksquare$