Class CBSE Class 12 Mathematics Differential Equations Q #1849
COMPETENCY BASED
ANALYZE
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
The order and degree of the differential equation $\frac{d}{dx}(e^y) = 0$ respectively are
(A) 0, 1
(B) 1, 1
(C) 2, 1
(D) 1, not defined
Prev Finish
Correct Answer: B

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Detailed Solution

Step 1: Simplify the given differential equation

The given equation is d/dx(e^y) = 0. Using the chain rule of differentiation, we differentiate e^y with respect to x:

$$e^y \cdot \frac{dy}{dx} = 0$$

Step 2: Identify the order

The order of a differential equation is the order of the highest derivative present in the equation. Here, the highest derivative is dy/dx, which is a first-order derivative. Thus, the order is 1.

Step 3: Identify the degree

The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in terms of derivatives. In the equation e^y \cdot (dy/dx) = 0, the term e^y is not a polynomial in terms of dy/dx. However, since e^y is never zero, we can divide by it to get dy/dx = 0. This is a polynomial in dy/dx with power 1. Therefore, the degree is 1.

Final Answer: 1, 1

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Pedagogical Audit
Bloom's Analysis: This is an ANALYZE question because the student must first simplify the expression using calculus rules before identifying the properties of the differential equation.
Knowledge Dimension: PROCEDURAL
Justification: The student must apply the chain rule procedure to transform the equation into a standard form before determining order and degree.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This tests the conceptual clarity of differential equations beyond rote memorization of definitions.

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