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The given equation is a second-order linear non-homogeneous differential equation:

\[
y'' - 6y' + 9y = 9x^2 - 12x + 2
\]

Let's solve it step by step:

### 1. Solve the homogeneous equation:
The homogeneous equation is:
\[
y'' - 6y' + 9y = 0
\]

This can be rewritten as:
\[
(r^2 - 6r + 9) = 0
\]

Factorize the quadratic:
\[
(r - 3)^2 = 0
\]

Thus, \( r = 3 \) is a repeated root. The general solution of the homogeneous equation is:
\[
y_h = C_1 e^{3x} + C_2 x e^{3x}
\]

### 2. Find the particular solution (\( y_p \)):
The right-hand side is a quadratic polynomial \( 9x^2 - 12x + 2 \). Assume a particular solution of the form:
\[
y_p = Ax^2 + Bx + C
\]

Compute \( y_p' \) and \( y_p'' \):
\[
y_p' = 2Ax + B, \quad y_p'' = 2A
\]

Substitute \( y_p, y_p', y_p'' \) into the original equation:
\[
(2A) - 6(2Ax + B) + 9(Ax^2 + Bx + C) = 9x^2 - 12x + 2
\]

Expand:


Learn how to solve a second-order linear non-homogeneous differential equation. This problem covers methods such as solving the homogeneous part, finding particular solutions using undetermined coefficients, and applying superposition. Suitable for students in Grades 11-12 or those studying introductory college-level differential equations.

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