Math Problem Statement

y'' - 6y' + 9y = 9x^2 - 12x + 2

Solution

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

y6y+9y=9x212x+2y'' - 6y' + 9y = 9x^2 - 12x + 2

Let's solve it step by step:

1. Solve the homogeneous equation:

The homogeneous equation is: y6y+9y=0y'' - 6y' + 9y = 0

This can be rewritten as: (r26r+9)=0(r^2 - 6r + 9) = 0

Factorize the quadratic: (r3)2=0(r - 3)^2 = 0

Thus, r=3r = 3 is a repeated root. The general solution of the homogeneous equation is: yh=C1e3x+C2xe3xy_h = C_1 e^{3x} + C_2 x e^{3x}

2. Find the particular solution (ypy_p):

The right-hand side is a quadratic polynomial 9x212x+29x^2 - 12x + 2. Assume a particular solution of the form: yp=Ax2+Bx+Cy_p = Ax^2 + Bx + C

Compute ypy_p' and ypy_p'': yp=2Ax+B,yp=2Ay_p' = 2Ax + B, \quad y_p'' = 2A

Substitute yp,yp,ypy_p, y_p', y_p'' into the original equation: (2A)6(2Ax+B)+9(Ax2+Bx+C)=9x212x+2(2A) - 6(2Ax + B) + 9(Ax^2 + Bx + C) = 9x^2 - 12x + 2

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Second-Order Linear Equations
Non-Homogeneous Equations
Particular Solutions

Formulas

Homogeneous Equation Solution
Particular Solution Assumption

Theorems

Method of Undetermined Coefficients
Superposition Principle

Suitable Grade Level

Grades 11-12, College-level