First-Order Linear Homogeneous Differential Equations with Constant Coefficients
Theorem
The general solution to a first-order linear homogeneous differential equation with constant coefficient
$$ \frac{ d y}{ d x}=\alpha y $$
is
$$ y=Ae^{\alpha x} $$
where $A$ is a constant.
Description
Thinking about what function is equal to its derivative helps to understand why the exponential function is the answer.
Proof
By separating the variables in
$$ \frac{ d y}{ d x}=\alpha y $$,
and integrating both sides, we get
$$ \ln y=ax+C $$
where $C$ is an integration constant. Stripping the logarithm from $y$,
$$ y=e^{\alpha x + C}=e^{\alpha x} e^{C}=Ae^{\alpha x} $$
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