Class CBSE Class 12 Mathematics Differential Equations Q #651
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
The solution for the differential equation \(\log(\frac{dy}{dx})=3x+4y\) is:
(A) \(3e^{4y}+4e^{-3x}+C=0\)
(B) \(e^{3x+4y}+C=0\)
(C) \(3e^{-3y}+4e^{4x}+12C=0\)
(D) \(3e^{-4y}+4e^{3x}+12C=0\)
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Correct Answer: D

AI Tutor Explanation

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Step-by-Step Solution

  1. Given the differential equation: \(\log(\frac{dy}{dx}) = 3x + 4y\)

  2. Exponentiate both sides to remove the logarithm: \(\frac{dy}{dx} = e^{3x + 4y}\)

  3. Rewrite the exponential term: \(\frac{dy}{dx} = e^{3x} \cdot e^{4y}\)

  4. Separate the variables: \(e^{-4y} dy = e^{3x} dx\)

  5. Integrate both sides: \(\int e^{-4y} dy = \int e^{3x} dx\)

  6. Perform the integration: \(-\frac{1}{4} e^{-4y} = \frac{1}{3} e^{3x} + C\)

  7. Multiply both sides by -12 to eliminate fractions: \(3e^{-4y} = -4e^{3x} - 12C\)

  8. Rearrange the equation: \(3e^{-4y} + 4e^{3x} + 12C = 0\)

Correct Answer: \(3e^{-4y}+4e^{3x}+12C=0\)<\/strong>

AI Suggestion: Option D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of differential equations and integration to find the solution.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to solve the differential equation, including separation of variables and integration.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge and application of standard methods for solving differential equations, a topic covered in the textbook.

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