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

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Question: Find the general solution of 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, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).

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

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

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

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: Since we have two distinct real roots (2 and 3), the general solution of the differential equation 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 a second-order linear homogeneous differential equation with constant coefficients.
Characteristic Equation – It assesses the understanding of deriving the characteristic equation from the differential equation and finding its roots.
General Solution – The question evaluates the ability to write the general solution based on the roots of the characteristic equation.