Question

The integrating factor of the differential equation \( x \frac{dy}{dx} - y = x^4 - 3x \) is:

• A. \( x \)
• B. \( -x \)
• C. \( x^{-1} \)
• D. \( \log(x^{-1}) \)
• E.

Hint:

The integrating factor for a linear differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \) is given by \( e^{\int P(x) \, dx} \). Simplify the exponent carefully to find the IF.

Answer 1

The Correct Option is C

The differential equation given is: \[ x \frac{dy}{dx} - y = x^4 - 3x. \] In standard linear form, this becomes: \[ \frac{dy}{dx} - \frac{y}{x} = x^3 - \frac{3}{x}. \] The coefficient of \( y \) is \( P(x) = -\frac{1}{x} \). The integrating factor (IF) is calculated using the formula: \[ \text{Integrating Factor} = e^{\int P(x) \, dx}. \] Substituting \( P(x) \): \[ \text{IF} = e^{\int -\frac{1}{x} \, dx}. \] The integral of \( -\frac{1}{x} \) is: \[ \int -\frac{1}{x} \, dx = -\ln|x|. \] Therefore: \[ \text{IF} = e^{-\ln|x|}. \] Using the property \( e^{\ln a} = a \), this simplifies to: \[ \text{IF} = |x|^{-1}. \] For positive \( x \), the integrating factor is: \[ \text{IF} = x^{-1}. \] Thus, the integrating factor is \( x^{-1} \).