Solve the differential equation dy/dx + 2y = 4.

Solve the differential equation dy/dx + 2y = 4.

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

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