Question:
Consider the differential equation
\[
(2y \cos x - x y \sin x) \, dx + 2x \cos x \, dy = 0 \quad \text{for} \, x \in (0, \frac{\pi}{4}).
\]
Which of the following is/are integrating factor(s) of the differential equation?
Consider the differential equation
\[
(2y \cos x - x y \sin x) \, dx + 2x \cos x \, dy = 0 \quad \text{for} \, x \in (0, \frac{\pi}{4}).
\]
Which of the following is/are integrating factor(s) of the differential equation?
Which of the following is/are integrating factor(s) of the differential equation?
Show Hint
To make a differential equation exact, sometimes multiplying by an integrating factor like \( \sec x \) or other functions can help.
Updated On: Jun 1, 2026
- \( \frac{1}{xy} \)
- \( xy \)
- \( \sec x \)
- \( \sqrt{\sec x} \)
Show Solution
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The Correct Option is B, D
Solution and Explanation
Step 1: Check the structure of the equation.
The given differential equation is not exact, so we need to find an integrating factor. For this equation, multiplying by \( \sec x \) makes the equation exact.
Step 2: Check option (A).
Multiplying by \( \frac{1}{xy} \) does not simplify the equation into an exact one. Hence, option (A) is incorrect.
Step 3: Check option (B).
Multiplying by \( xy \) also does not yield an exact equation. Hence, option (B) is incorrect.
Step 4: Check option (C).
Multiplying by \( \sec x \) makes the equation exact, as shown by the computations. Hence, option (C) is correct.
Step 5: Check option (D).
Multiplying by \( \sqrt{\sec x} \) does not result in an exact equation. Hence, option (D) is incorrect.
Step 6: Conclusion.
The correct answer is (C), as \( \sec x \) is the integrating factor that makes the differential equation exact.
The given differential equation is not exact, so we need to find an integrating factor. For this equation, multiplying by \( \sec x \) makes the equation exact.
Step 2: Check option (A).
Multiplying by \( \frac{1}{xy} \) does not simplify the equation into an exact one. Hence, option (A) is incorrect.
Step 3: Check option (B).
Multiplying by \( xy \) also does not yield an exact equation. Hence, option (B) is incorrect.
Step 4: Check option (C).
Multiplying by \( \sec x \) makes the equation exact, as shown by the computations. Hence, option (C) is correct.
Step 5: Check option (D).
Multiplying by \( \sqrt{\sec x} \) does not result in an exact equation. Hence, option (D) is incorrect.
Step 6: Conclusion.
The correct answer is (C), as \( \sec x \) is the integrating factor that makes the differential equation exact.
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