Linear Second Order ODE/y'' - 4 y = 0
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Theorem
The second order ODE:
- $(1): \quad y'' - 4 y = 0$
has the general solution:
- $y = C_1 e^{2 x} + C_2 e^{-2 x}$
Proof 1
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
- $(2): \quad: m^2 - 4 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
- $m_1 = 2$
- $m_2 = -2$
These are complex and unequal.
So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
- $y = C_1 e^{2 x} + C_2 e^{-2 x}$
$\blacksquare$
Proof 2
This is an instance of:
which yields:
- $y = C_1 e^{k x} + C_2 e^{-k x}$
where $k = 2$.
Hence the result.
$\blacksquare$