Solve the equation y' = 6y + 12.

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Question: Solve the equation y\' = 6y + 12.

Options:

Correct Answer: y = 2 - Ce^(-6x)

Solution:

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

Solve the equation y' = 6y + 12.

Solve the equation y' = 6y + 12.

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.

First-Order Linear Differential Equations – These are equations of the form y' + P(x)y = Q(x), which can be solved using an integrating factor.
Integrating Factor Method – A technique used to solve first-order linear differential equations by multiplying through by a function that simplifies the equation.
Exponential Functions – The solution involves exponential functions, particularly e raised to a power, which is common in solutions to differential equations.