Linear Second Order ODE/y'' + 2 y' + y = 0

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