Solve the differential equation y'' - 3y' + 2y = 0.

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Question: Solve the differential equation y\'\' - 3y\' + 2y = 0.

Options:

Correct Answer: y = C1e^(x) + C2e^(2x)

Solution:

The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).

Solve the differential equation y'' - 3y' + 2y = 0.

Solve the differential equation y'' - 3y' + 2y = 0.

Step 1: Identify the given differential equation: y'' - 3y' + 2y = 0.
Step 2: Write the characteristic equation by replacing y'' with r^2, y' with r, and y with 1: r^2 - 3r + 2 = 0.
Step 3: Factor the characteristic equation: (r - 1)(r - 2) = 0.
Step 4: Solve for r by setting each factor to zero: r - 1 = 0 gives r = 1, and r - 2 = 0 gives r = 2.
Step 5: Write the general solution using the values of r: y = C1 * e^(1x) + C2 * e^(2x), where C1 and C2 are constants.

Homogeneous Linear Differential Equations – The question tests the ability to solve second-order homogeneous linear differential equations with constant coefficients.
Characteristic Equation – It assesses the understanding of deriving the characteristic equation from the differential equation and solving it for roots.
General Solution – The question evaluates the ability to construct the general solution from the roots of the characteristic equation.