Question

Find the integrating factor (I.F.) for the linear differential equation: \[ \frac{dy}{dx} - y\tan x = e^x \]

• A. \( \cos x \)
• B. \( \sec x \)
• C. \( -\cos x \)
• D. \( \ln|\cos x| \)

Hint:

Remember to carry the negative sign along with your \( P(x) \) function. Missing the negative sign will lead you to calculate \( \sec x \) instead of the correct answer, which is a common multiple-choice trap.

Answer 1

The Correct Option is A

Concept: For a linear differential equation written in the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \), the Integrating Factor is calculated using the formula: \[ \text{I.F.} = e^{\int P(x)\,dx} \] Be sure to include any negative signs when identifying the function \( P(x) \).

Step 1: Isolate the function \( P(x) \) with its proper sign. By comparing our given equation to the standard layout, the coefficient function of \( y \) includes the negative sign: \[ P(x) = -\tan x \]

Step 2: Integrate the function \( P(x) \) with respect to \( x \). Evaluate the indefinite integral of the negative tangent function component: \[ \int P(x)\,dx = \int -\tan x\,dx = -\int \tan x\,dx \] Using standard trigonometric integration rules, we know that \( \int \tan x\,dx = \ln|\sec x| \): \[ -\int \tan x\,dx = -\ln|\sec x| \] Using logarithmic properties to simplify the negative coefficient into an exponent: \[ -\ln|\sec x| = \ln\left|(\sec x)^{-1}\right| = \ln\left|\frac{1}{\sec x}\right| = \ln|\cos x| \]

Step 3: Evaluate the exponential expression to find the I.F. Plugging this result back into our exponential base formula: \[ \text{I.F.} = e^{\ln|\cos x|} = \cos x \]