Solve the given Differential Equation by Undetermined Coefficients. 1/4 y'' + y' + y = x² − 4x

Solution:

We will be solving this by finding the homogeneous and complementary solutions of the equation. Let's solve this step by step.

Given, differential equation: 1/4 y'' + y' + y = x² − 4x

First solve the corresponding homogeneous differential equation: 1/4 y'' + y' + y = 0

The characteristic equation is 1/4 r² + r + 1 = 0

Let's find its roots by factorization.

(1/2 r + 1) (1/2 r + 1) = 0

1/2 r + 1 = 0

r = -2

Therefore, r = -2 is a repeated root thus one of the complimentary solutions y_c is y_c = c₁e^-2x + c₂xe^-2x 

Now, find the remaining complimentary solution y_p.

y_p = Ax² + Bx + C

y′_p = 2Ax + B

y′′_p = 2A

Substitute y_p, y′_p, and y′′_p in given differential equation and solve for A and B.

1/4(2A) + (2Ax + B) + (Ax² + Bx + C) = x² − 4x

1/2A + 2Ax + B + Ax² + Bx +C = x² − 4x

Ax² + x(2A + B) + (1/2A + B + C) = x² − 4x

On comparing, we get A = 1, 2A + B = -4, 1/2A + B + C = 0

So, B = -6

and,

1/2A + B + C = 0

1/2 - 6 + C = 0

C = 5 -1/2

C = 11/2

Therefore, y_p = x²  - 6x + 11/2

Now, combine complementary solution and particular solution together to arrive at the general solution.

y = y_c + y_p 

y = c₁e^-2x + c₂xe^-2x + x²  - 6x + 11/2

Thus, the general solution of differential equation 1/4 y'' + y' + y = x² − 3x is y = c₁e^-2x + c₂xe^-2x + x² - 6x + 11/2

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Solve the given Differential Equation by Undetermined Coefficients. 1/4 y'' + y' + y = x² − 4x

Summary:

The general solution of differential equation 1/4 y'' + y' + y = x² − 4x is y = c₁e^-2x + c₂xe^-2x + x² - 6x + 11/2