Find the general solution of the equation y'' - 5y' + 6y = 0.

Find the general solution of the equation y'' - 5y' + 6y = 0.

Step 1: Write down the given differential equation: y'' - 5y' + 6y = 0.
Step 2: Identify the characteristic equation by replacing y'' with r^2, y' with r, and y with 1. This gives us r^2 - 5r + 6 = 0.
Step 3: Factor the characteristic equation r^2 - 5r + 6. Look for two numbers that multiply to 6 and add to -5. The numbers are -2 and -3.
Step 4: Write the factored form: (r - 2)(r - 3) = 0.
Step 5: Set each factor equal to zero to find the roots: r - 2 = 0 gives r = 2, and r - 3 = 0 gives r = 3.
Step 6: Write the general solution using the roots found. The general solution is y = C1 e^(2x) + C2 e^(3x), where C1 and C2 are constants.

Homogeneous Linear Differential Equations – The question tests the ability to solve second-order homogeneous linear differential equations using characteristic equations.
Characteristic Equation – Understanding how to derive and solve the characteristic equation from the given differential equation.
General Solution – The requirement to express the general solution in terms of arbitrary constants after finding the roots.