Linearly Independent Solutions of y'' - y = 0

Theorem
The second order ODE:
 * $(1): \quad y'' - y = 0$

has particular solutions:
 * $y_1 = e^x$
 * $y_2 = e^{-x}$

which are linearly independent.

Proof
We have that:

Hence it can be seen by inspection that:

are particular solutions to $(1)$.

Calculating the Wronskian of $y_1$ and $y_2$:

So the Wronskian of $y_1$ and $y_2$ is never zero.

Thus from Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent:
 * $y_1$ and $y_2$ are linearly independent.