Linear Second Order ODE/y'' - 2 y' + y = 2 x/Proof 3

Proof
It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
 * $y'' + p y' + q y = R \left({x}\right)$

where:
 * $p = -2$
 * $q = 1$
 * $R \left({x}\right) = 2 x$

First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
 * $(2): \quad y'' - 2 y' + y = 0$

From Second Order ODE: $y'' - 2 y' + y = 0$, this has the general solution:
 * $y_g = C_1 e^x + C_2 x e^x$

We have that:
 * $R \left({x}\right) = 2 x$

So from the Method of Undetermined Coefficients for Polynomial:
 * $y_p = A_0 + A_1 x$

Hence:

Substituting into $(1)$:

So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:


 * $y = y_g + y_p = C_1 e^x + C_2 x e^x + 2 x + 4$

is the general solution to $(1)$.