Linear Second Order ODE/2 y'' + 2 y' + 3 y = 0
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Theorem
The second order ODE:
- $(1): \quad 2 y + 2 y' + 3 y = 0$
has the general solution:
- $y = e^{-x/2} \paren {C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}$
Proof
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
- $(2): \quad: 2 m^2 + 2 m + 3 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
- $m_1 = -\dfrac 1 2 + \dfrac {\sqrt 5} 2 i$
- $m_2 = -\dfrac 1 2 - \dfrac {\sqrt 5} 2 i$
These are complex and unequal.
So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
- $y = e^{-x/2} \paren {C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}$
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.17$: Problem $1 \ \text{(g)}$