What is the integrating factor for the equation dy/dx + (1/x)y = 2?

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Question: What is the integrating factor for the equation dy/dx + (1/x)y = 2?

Options:

Correct Answer: e^(ln(x^2))

Solution:

The integrating factor is e^(∫(1/x)dx) = e^(ln(x)) = x.

What is the integrating factor for the equation dy/dx + (1/x)y = 2?

What is the integrating factor for the equation dy/dx + (1/x)y = 2?

Step 1: Identify the equation you have, which is dy/dx + (1/x)y = 2.
Step 2: Recognize that this is a first-order linear differential equation.
Step 3: Find the coefficient of y, which is (1/x).
Step 4: To find the integrating factor, you need to calculate e^(∫(1/x)dx).
Step 5: Calculate the integral ∫(1/x)dx, which equals ln(x).
Step 6: Substitute the result of the integral back into the expression for the integrating factor: e^(ln(x)).
Step 7: Simplify e^(ln(x)), which equals x.
Step 8: Conclude that the integrating factor is x.

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