Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation

Theorem
Let $p^2 > 4 q$.

Then $(1)$ has the general solution:


 * $y = C_1 e^{m_1 x} + C_2 e^{m_2 x}$

Proof
Consider the auxiliary equation of $(1)$:
 * $(2): \quad m^2 + p m + q$

Let $p^2 > 4 q$.

From Solution to Quadratic Equation: Real Coefficients, $(2)$ has two real roots:

As $p^2 > 4 q$ we have that:
 * $\sqrt {\dfrac {p^2} 4 - q} \ne 0$

and so:
 * $m_1 \ne m_2$

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

are both particular solutions to $(1)$.

We also have that:

Thus $y_1$ and $y_2$ are linearly independent.

It follows from Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution that:
 * $y = C_1 e^{m_1 x} + C_2 e^{m_2 x}$

is the general solution to $(1)$.