What is the integrating factor for the equation dy/dx + 3y = 6x?

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Question: What is the integrating factor for the equation dy/dx + 3y = 6x?

Options:

Correct Answer: e^(3x)

Solution:

The integrating factor is e^(∫3dx) = e^(3x).

What is the integrating factor for the equation dy/dx + 3y = 6x?

What is the integrating factor for the equation dy/dx + 3y = 6x?

Step 1: Identify the standard form of the differential equation, which is dy/dx + P(x)y = Q(x). In this case, P(x) = 3 and Q(x) = 6x.
Step 2: Find the integrating factor, which is given by the formula e^(∫P(x)dx). Here, P(x) = 3.
Step 3: Calculate the integral of P(x): ∫3dx = 3x.
Step 4: Substitute the result of the integral into the formula for the integrating factor: e^(∫3dx) = e^(3x).
Step 5: Conclude that the integrating factor for the equation dy/dx + 3y = 6x is e^(3x).

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