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 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:

$y = C_1 e^x + C_2 x e^x$

$\blacksquare$