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:

Linear Second Order ODE: $y'' - k^2 y = 0$

which yields:

$y = C_1 e^{k x} + C_2 e^{-k x}$

where $k = 2$.

Hence the result.

$\blacksquare$