Solve the first-order linear differential equation dy/dx + 2y = 6.

Solve the first-order linear differential equation dy/dx + 2y = 6.

Step 1: Write down the differential equation: dy/dx + 2y = 6.
Step 2: Identify the integrating factor. The integrating factor is e^(∫2dx) = e^(2x).
Step 3: Multiply the entire equation by the integrating factor: e^(2x) * (dy/dx) + 2e^(2x) * y = 6e^(2x).
Step 4: The left side of the equation can be rewritten as the derivative of a product: d/dx(e^(2x) * y) = 6e^(2x).
Step 5: Integrate both sides with respect to x: ∫d/dx(e^(2x) * y) dx = ∫6e^(2x) dx.
Step 6: The left side simplifies to e^(2x) * y. For the right side, the integral of 6e^(2x) is 3e^(2x) + C, where C is the constant of integration.
Step 7: Set the two sides equal: e^(2x) * y = 3e^(2x) + C.
Step 8: Solve for y by dividing both sides by e^(2x): y = 3 + Ce^(-2x).
Step 9: Rearrange the equation to match the final form: y = 3 - Ce^(-2x).

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