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 given differential equation is: \[ x \frac{dy}{dx} - y = x^4 - 3x. \] This can be rewritten in standard linear form: \[ \frac{dy}{dx} - \frac{y}{x} = x^3 - \frac{3}{x}. \] Here, the coefficient of \( y \) is \( -\frac{1}{x} \). To find the integrating factor (IF), use the formula: \[ \text{Integrating Factor} = e^{\int P(x) \, dx}, \] where \( P(x) = -\frac{1}{x} \). Thus: \[ \text{IF} = e^{\int -\frac{1}{x} \, dx}. \] Integrating \( -\frac{1}{x} \): \[ \int -\frac{1}{x} \, dx = -\ln|x|. \] So: \[ \text{IF} = e^{-\ln|x|}. \] Using the property \( e^{\ln a} = a \), this simplifies to: \[ \text{IF} = |x|^{-1}. \] For positive \( x \), we can write: \[ \text{IF} = x^{-1}. \] Hence, the integrating factor is (C) \( x^{-1} \).