Linear Second Order ODE/2 y'' - 4 y' + 8 y = 0
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Theorem
The second order ODE:
- $(1): \quad 2 y - 4 y + 8 y = 0$
has the general solution:
- $y = e^x \paren {C_1 \cos \sqrt 3 x + C_2 \sin \sqrt 3 x}$
Proof
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Let $(1)$ be written in the form:
- $y - 2 y + 4 y = 0$
Its auxiliary equation is:
- $(2): \quad: m^2 - 2 m + 4 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
- $m_1 = 1 + \sqrt 3 i$
- $m_2 = 1 - \sqrt 3 i$
These are complex and unequal.
So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
- $y = e^x \paren {C_1 \cos \sqrt 3 x + C_2 \sin \sqrt 3 x}$
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.17$: Problem $1 \ \text{(d)}$