What is the solution of the equation dy/dx = 4y + 2? (2021)

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Question: What is the solution of the equation dy/dx = 4y + 2? (2021)

Options:

Correct Answer: y = Ce^(4x) - 1/2

Exam Year: 2021

Solution:

Using an integrating factor, the solution is y = Ce^(4x) - 1/2.

What is the solution of the equation dy/dx = 4y + 2? (2021)

What is the solution of the equation dy/dx = 4y + 2? (2021)

Step 1: Identify the equation dy/dx = 4y + 2. This is a first-order linear differential equation.
Step 2: Rewrite the equation in standard form: dy/dx - 4y = 2.
Step 3: Find the integrating factor, which is e^(∫-4dx) = e^(-4x).
Step 4: Multiply the entire equation by the integrating factor: e^(-4x) * dy/dx - 4e^(-4x) * y = 2e^(-4x).
Step 5: The left side of the equation can be rewritten as the derivative of a product: d/dx (e^(-4x) * y) = 2e^(-4x).
Step 6: Integrate both sides: ∫d/dx (e^(-4x) * y) dx = ∫2e^(-4x) dx.
Step 7: The left side simplifies to e^(-4x) * y. For the right side, the integral is -1/2 * e^(-4x) + C (where C is the constant of integration).
Step 8: Set the equation: e^(-4x) * y = -1/2 * e^(-4x) + C.
Step 9: Multiply both sides by e^(4x) to solve for y: y = Ce^(4x) - 1/2.

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