Question

The integrating factor of the differential equation \[ e^{-\frac{2x}{\sqrt{x}}} \, \frac{dy}{dx} = 1 \] is:

• A. \( e^{-1/\sqrt{x}} \)
• B. \( e^{2/\sqrt{x}} \)
• C. \( e^{2x} \)
• D. \( e^{-2x} \)

Hint:

To find the integrating factor, identify the function \( P(x) \) in the differential equation and integrate it.

Answer 1

The Correct Option is A

The integrating factor for a first-order linear differential equation of the form \( \frac{dy}{dx} + P(x) y = Q(x) \) is given by: \[ \mu(x) = e^{\int P(x) \, dx}. \] In this case, the integrating factor for \( e^{-\frac{2x}{\sqrt{x}}} \frac{dy}{dx} = 1 \) is \( \mu(x) = e^{-1/\sqrt{x}} \).