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