What is the solution to the differential equation dy/dx = 2y + 3?

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Question: What is the solution to the differential equation dy/dx = 2y + 3?

Options:

Correct Answer: y = Ce^(2x) + 3/2

Solution:

Using the integrating factor method, we find y = Ce^(2x) + 3/2.

What is the solution to the differential equation dy/dx = 2y + 3?

What is the solution to the differential equation dy/dx = 2y + 3?

Step 1: Start with the differential equation dy/dx = 2y + 3.
Step 2: Rewrite the equation in standard form: dy/dx - 2y = 3.
Step 3: Identify the integrating factor, which is e^(∫-2dx) = e^(-2x).
Step 4: Multiply the entire equation by the integrating factor: e^(-2x) * dy/dx - 2e^(-2x) * y = 3e^(-2x).
Step 5: The left side of the equation can be rewritten as the derivative of (e^(-2x) * y).
Step 6: So, we have d/dx(e^(-2x) * y) = 3e^(-2x).
Step 7: Integrate both sides with respect to x: ∫d/dx(e^(-2x) * y) dx = ∫3e^(-2x) dx.
Step 8: The left side simplifies to e^(-2x) * y, and the right side integrates to -3/2 * e^(-2x) + C.
Step 9: Now, we have e^(-2x) * y = -3/2 * e^(-2x) + C.
Step 10: Multiply both sides by e^(2x) to solve for y: y = Ce^(2x) - 3/2.

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