Class CBSE Class 12 Mathematics Differential Equations Q #1719
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
$\frac{dy}{dx}=F(x,y)$ will be a homogeneous differential equation for which of the following functions? (i) $F(x,y)=3x+2y$ (ii) $F(x,y)=\sin\frac{y}{x}+\log y-\log x$ (iii) $F(x,y)=e^{y/x}+1$ (iv) $F(x,y)=\sqrt{x^{2}+y^{2}}-y$
(A) (i) and (ii)
(B) (i), (ii) and (iii)
(C) (ii), (iii) and (iv)
(D) (ii) and (iii)
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Correct Answer: D

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Detailed Solution

Step 1: Definition of Homogeneous Differential Equation

A differential equation of the form dy/dx = F(x, y) is homogeneous if the function F(x, y) is a homogeneous function of degree zero. This means F(λx, λy) = λ0 F(x, y) = F(x, y).

Step 2: Testing each function

(i) F(x, y) = 3x + 2y: F(λx, λy) = 3(λx) + 2(λy) = λ(3x + 2y). Since this is λ1F(x, y), it is not homogeneous of degree zero.
(ii) F(x, y) = sin(y/x) + log(y) - log(x) = sin(y/x) + log(y/x): F(λx, λy) = sin(λy/λx) + log(λy/λx) = sin(y/x) + log(y/x). This is homogeneous of degree zero.
(iii) F(x, y) = ey/x + 1: F(λx, λy) = eλy/λx + 1 = ey/x + 1. This is homogeneous of degree zero.
(iv) F(x, y) = √(x2 + y2) - y: F(λx, λy) = √(λ2x2 + λ2y2) - λy = λ√(x2 + y2) - λy = λ(√(x2 + y2) - y). This is homogeneous of degree one, not zero.

Step 3: Conclusion

Functions (ii) and (iii) satisfy the condition for a homogeneous differential equation.

Final Answer: (D)

AI Suggestion: Option D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the mathematical definition of a homogeneous function to verify specific algebraic expressions.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the underlying concept of homogeneity in differential equations rather than rote memorization of formulas.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It requires the student to evaluate multiple cases to identify which functions satisfy the criteria for a specific mathematical classification.

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