Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation

Theorem
Let:
 * $(1): \quad y'' + p y' + q y = 0$

be a constant coefficient homogeneous linear second order ODE.

Then:
 * $y = e^{m_1 x}$ is a solution to $(1)$


 * $m_1$ is a root of the auxiliary equation $m^2 + p m + q = 0$

Proof
Consider the equation:
 * $y = e^{m_1 x}$

Differentiating $x$:

Sufficient Condition
Let $y = e^{m_1 x}$ be a solution to $(1)$.

Substituting for $y$ and its derivatives in $(1)$:

That is, by definition, $m_1$ is a root of $m^2 + p m + q = 0$.

Necessary Condition
Let $m_1$ be a root of $m^2 + p m + q = 0$.

Thus:

Thus $y = e^{m_1 x}$ satisfies $(1)$.