What is the solution of the equation y'' - 4y' + 4y = 0?

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Question: What is the solution of the equation y\'\' - 4y\' + 4y = 0?

Options:

Correct Answer: y = (C1 + C2x)e^(2x)

Solution:

The characteristic equation has a repeated root, leading to the solution form (C1 + C2x)e^(2x).

What is the solution of the equation y'' - 4y' + 4y = 0?

What is the solution of the equation y'' - 4y' + 4y = 0?

Step 1: Write down the given differential equation: y'' - 4y' + 4y = 0.
Step 2: Identify the characteristic equation by replacing y'' with r^2, y' with r, and y with 1: r^2 - 4r + 4 = 0.
Step 3: Factor the characteristic equation: (r - 2)(r - 2) = 0.
Step 4: Solve for r: The repeated root is r = 2.
Step 5: Write the general solution for a repeated root: y = (C1 + C2x)e^(rx), where r is the repeated root.
Step 6: Substitute r = 2 into the general solution: y = (C1 + C2x)e^(2x).

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