Class CBSE Class 12 Mathematics Differential Equations Q #1718
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
Which of the following is not a Linear Differential Equation?
(A) $(1+x^{2})dy+2xy~dx=\cot x~dx$
(B) $y+\frac{d}{dx}(xy)=x(\sin x+\log x)$
(C) $x(1+y^{2})dx-y(1+x^{2})dy=0$
(D) $y~dx-(x+3y^{2})dy=0$
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Correct Answer: C

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Detailed Solution

Step 1: Understanding the Definition

A linear differential equation of the first order is of the form dy/dx + Py = Q or dx/dy + Px = Q, where P and Q are constants or functions of x (or y respectively). The dependent variable and its derivative must appear only in the first degree and not be multiplied together.

Step 2: Analyzing Option (A)

Rearranging (1+x^2)dy + 2xy dx = cot x dx gives dy/dx + (2x/(1+x^2))y = cot x / (1+x^2). This is linear in y.

Step 3: Analyzing Option (B)

Expanding y + d/dx(xy) = x(sin x + log x) gives y + y + x(dy/dx) = x(sin x + log x), which simplifies to x(dy/dx) + 2y = x(sin x + log x). This is linear in y.

Step 4: Analyzing Option (C)

Rearranging x(1+y^2)dx - y(1+x^2)dy = 0 gives dx/dy = y(1+x^2) / (x(1+y^2)). This is a variable separable equation, but it is not linear because the term x^2 appears in the numerator, preventing the form dx/dy + Px = Q.

Step 5: Analyzing Option (D)

Rearranging y dx - (x + 3y^2) dy = 0 gives dx/dy = (x + 3y^2) / y, which simplifies to dx/dy - (1/y)x = 3y. This is linear in x.

Final Answer: C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the standard definition of linear differential equations to identify non-conforming forms.
Knowledge Dimension: PROCEDURAL
Justification: The student must perform algebraic manipulation and rearrangement to test the linearity condition.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the conceptual clarity of the chapter 'Differential Equations' beyond rote memorization.

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