Solve the equation y' = 6y + 12.

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Q: Solve the equation y' = 6y + 12.

Solution: This is a first-order linear equation. The integrating factor method gives the solution y = 2 - Ce^(-6x).

Steps: 10

Step 1: Identify the equation. We have y' = 6y + 12.
Step 2: Rewrite the equation in standard form. This means we want it to look like y' - 6y = 12.
Step 3: Identify the integrating factor. The integrating factor is e^(∫-6 dx) = e^(-6x).
Step 4: Multiply the entire equation by the integrating factor. This gives us e^(-6x) * y' - 6e^(-6x) * y = 12e^(-6x).
Step 5: Recognize the left side as the derivative of a product. The left side can be written as d/dx(e^(-6x) * y).
Step 6: Integrate both sides. ∫d/dx(e^(-6x) * y) dx = ∫12e^(-6x) dx.
Step 7: Solve the integral on the right side. The integral of 12e^(-6x) is -2e^(-6x) + C, where C is the constant of integration.
Step 8: Set the left side equal to the right side. e^(-6x) * y = -2e^(-6x) + C.
Step 9: Solve for y by multiplying both sides by e^(6x). This gives us y = -2 + Ce^(6x).
Step 10: Rearrange the equation to match the short solution format. We can write it as y = 2 - Ce^(-6x) by adjusting the constant.