Question

Match List-I (Differential equations) with List-II (Order and Degree).

• A. (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
• B. (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
• C. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
• D. (A)-(I), (B)-(III), (C)-(IV), (D)-(II)

Hint:

If an integral appears in a differential equation, differentiate it first to find order and degree correctly.

Answer 1

The Correct Option is A

Concept:
• Order: Highest order derivative present in the differential equation.
• Degree: Highest power of highest order derivative after removing radicals/fractions. ---

Step 1: {Analyze (A)}
\[ \frac{d^2y}{dx^2}=1+\sqrt{\frac{dy}{dx}} \] - Highest derivative: $\frac{d^2y}{dx^2}$ → Order = 2 - RHS contains square root of $\frac{dy}{dx}$ but highest derivative still second order - Degree is power of highest derivative → appears as power 1 \[ (A) \rightarrow \text{Order }2,\ \text{Degree }1 \Rightarrow (I) \] ---

Step 2: {Analyze (B)}
\[ \frac{dy}{dx}+2\frac{dx}{dy}=x \] - Highest derivative is first order - After rearrangement, still first order - Degree = 1 \[ (B) \rightarrow \text{Order }1,\ \text{Degree }1 \Rightarrow (II) \] ---

Step 3: {Analyze (C)}
\[ y+2\frac{dy}{dx}=\int y\,dx \] Differentiate both sides to remove integral: \[ \frac{dy}{dx}+2\frac{d^2y}{dx^2}=y \] - Highest derivative: second order → Order = 2 - Highest power = 1 → Degree = 2 \[ (C) \rightarrow \text{Order }2,\ \text{Degree }2 \Rightarrow (IV) \] ---

Step 4: {Analyze (D)}
\[ \frac{dy}{dx}+y=\log x \] - Highest derivative: first order - Degree = 1 \[ (D) \rightarrow \text{Order }1,\ \text{Degree }1 \Rightarrow (II) \] --- Final Matching: \[ (A)-(I),\ (B)-(II),\ (C)-(IV),\ (D)-(II) \] ---