Solve the second-order differential equation y'' - 6y' + 9y = 0.

Step 1: Form the characteristic equation.

For:
y'' - 6y' + 9y = 0

replace the derivatives with powers of r:

r^2 - 6r + 9 = 0

Step 2: Factor the characteristic equation.

r^2 - 6r + 9 = (r - 3)^2

Therefore:
(r - 3)^2 = 0

Step 3: Identify the root.

The characteristic equation has the repeated root:
r = 3

This root has multiplicity 2.

Step 4: Recall the solution form for a repeated root.

If a second-order homogeneous equation has a repeated root r, two linearly independent solutions are:
e^(rx)
and
xe^(rx)

Step 5: Write the general solution.

Using r = 3:

y = C1e^(3x) + C2xe^(3x)

Step 6: Explain why the x factor is needed.

Writing C1e^(3x) + C2e^(3x) would not give two independent solutions because both terms would be multiples of the same function. Multiplying the second solution by x produces a second linearly independent solution.

Therefore, the general solution is y = C1e^(3x) + C2xe^(3x).