Solve the differential equation y' = 5 - 2y.

Solve the differential equation y' = 5 - 2y.

Step 1: Identify the differential equation. We have y' = 5 - 2y.
Step 2: Rewrite the equation in standard form. This means we want to isolate y' on one side: y' + 2y = 5.
Step 3: Identify the coefficients. Here, p(x) = 2 and q(x) = 5.
Step 4: Find the integrating factor. The integrating factor is e^(∫p(x)dx) = e^(∫2dx) = e^(2x).
Step 5: Multiply the entire equation by the integrating factor: e^(2x)y' + 2e^(2x)y = 5e^(2x).
Step 6: Recognize the left side as the derivative of a product: d/dx(e^(2x)y) = 5e^(2x).
Step 7: Integrate both sides with respect to x: ∫d/dx(e^(2x)y)dx = ∫5e^(2x)dx.
Step 8: The left side simplifies to e^(2x)y. For the right side, use integration: ∫5e^(2x)dx = (5/2)e^(2x) + C.
Step 9: Set the two sides equal: e^(2x)y = (5/2)e^(2x) + C.
Step 10: Solve for y by dividing both sides by e^(2x): y = (5/2) + Ce^(-2x).

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