Solve the second-order differential equation y'' - 5y' + 6y = 0.

Step 1: Recognize the equation type.

The equation:
y'' - 5y' + 6y = 0

is a second-order homogeneous linear differential equation with constant coefficients.

Step 2: Form the characteristic equation.

Replace y'' by r^2, y' by r, and y by 1:

r^2 - 5r + 6 = 0

Step 3: Factor the characteristic equation.

We need two numbers whose product is 6 and whose sum is -5.

Those numbers are -2 and -3.

Therefore:
(r - 2)(r - 3) = 0

Step 4: Find the characteristic roots.

r - 2 = 0 gives r = 2.

r - 3 = 0 gives r = 3.

The equation has two distinct real roots.

Step 5: Write the general solution.

For two distinct real roots r1 and r2, the general solution is:
y = C1e^(r1x) + C2e^(r2x)

Therefore:
y = C1e^(2x) + C2e^(3x)

Step 6: Explain the constants.

Because this is a second-order differential equation, the general solution contains two arbitrary constants, C1 and C2.

Therefore, the general solution is y = C1e^(2x) + C2e^(3x).