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

Step 1: Understanding the Concept:
This problem requires finding the Integrating Factor (I.F.) for a first-order linear differential equation.
The equation is given in the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \).
The purpose of the I.F. is to create an "exact derivative" on the left side of the equation.
Correct identification of \( P(x) \) is critical. It must include the sign associated with the term.
A common mistake in these problems is ignoring the negative sign preceding the \( y \) term, which leads to a completely different (and usually incorrect) answer.
Step 2: Key Formula or Approach:
The formula for the Integrating Factor is \( I.F. = e^{\int P(x) dx} \).
Relevant integral: \( \int \tan x dx = \ln|\sec x| \).
Relevant log property: \( - \ln A = \ln(A^{-1}) \).
Step 3: Detailed Explanation:
Starting with the given equation:
\[ \frac{dy}{dx} - y \tan x = e^x \]
Comparing this to the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \), we identify:
\[ P(x) = -\tan x \]
Now, we find the integral of \( P(x) \) with respect to \( x \):
\[ \int P(x) dx = \int (-\tan x) dx = -\int \tan x dx \]
We know that the integral of tangent is \( \ln|\sec x| \).
\[ \int P(x) dx = -\ln|\sec x| \]
Using the logarithmic identity where the coefficient becomes an exponent:
\[ -\ln|\sec x| = \ln|(\sec x)^{-1}| \]
Since the reciprocal of secant is cosine:
\[ \ln|(\sec x)^{-1}| = \ln\left|\frac{1}{\sec x}\right| = \ln|\cos x| \]
Now, apply the I.F. formula:
\[ \text{I.F.} = e^{\ln|\cos x|} \]
Using the identity \( e^{\ln f(x)} = f(x) \):
\[ \text{I.F.} = \cos x \]
Step 4: Final Answer:
The Integrating Factor is \( \cos x \).
This is provided in Option (A).