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

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Question: Solve the differential equation y\'\' - 5y\' + 6y = 0.

Options:

Correct Answer: y = C1 e^(3x) + C2 e^(2x)

Solution:

The characteristic equation is r^2 - 5r + 6 = 0, which factors to (r - 2)(r - 3) = 0, giving the solution y = C1 e^(2x) + C2 e^(3x).

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

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

Step 1: Identify the given differential equation: y'' - 5y' + 6y = 0.
Step 2: Write the characteristic equation by replacing y'' with r^2, y' with r, and y with 1: r^2 - 5r + 6 = 0.
Step 3: Factor the characteristic equation: (r - 2)(r - 3) = 0.
Step 4: Set each factor equal to zero to find the roots: r - 2 = 0 gives r = 2, and r - 3 = 0 gives r = 3.
Step 5: Write the general solution using the roots found: 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 – It assesses the understanding of deriving and solving the characteristic equation from the given differential equation.
Exponential Solutions – The solution involves recognizing that the roots of the characteristic equation lead to exponential functions in the general solution.