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Question #41584

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be a sufficiently differentiable function satisfying the equation $\lim_{t \to x} \frac{1}{\ln(1 + t - x)} \int_{x}^{t} \left( \frac{d^2y}{du^2} \right)^{3/5} du = \int_{0}^{x} y(v) \sqrt{v} \, dv$ for $x > 0$ . If the lowest-order pure differential equation (free from integrals) satisfied by $y(x)$ has order $O$ and degree $D$ , then the value of $O + D$ is:

Options

A.
8
B.
13
C.
5
D.
4

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Order and Degree Definition|Leibniz Rule for Differentiation under Integral|The L'Hôpital's Rule Misuse|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This problem tests the rigorous definition of the order and degree of a differential equation, combined with the Leibniz rule and L'Hôpital's rule.

The first trap lies in correctly evaluating the limit to form the integro-differential equation.

The second trap is recognizing that to find the order and degree of the pure differential equation, one must differentiate to eliminate the integral, shifting the order from 2 to 3.

The ultimate 'Advanced-level' trap exploits a common misconception regarding the degree: students often erroneously square the final equation to clear the fractional power on the independent variable $x$ (i.e., $x^{5/2}$ ), which incorrectly inflates the degree to 10.

The degree is strictly the power of the highest-order derivative when the equation is expressed as a polynomial in its derivatives only.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Order and Degree Definition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free