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.
$\blacksquare$