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Question #41585

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be a function satisfying the differential equation: $\frac{ \left( \frac{d^2y}{dx^2} \right)^2 - \left( \frac{dy}{dx} \right)^2 }{ \frac{d^2y}{dx^2} - \frac{dy}{dx} } = \frac{d^2y}{dx^2} + \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^3 - x$ If $O$ and $D$ represent the order and degree of this differential equation respectively, then the ordered pair $(O, D)$ is:

Options

A.
$(2, 1)$
B.
$(2, 2)$
C.
$(1, 3)$
D.
$(1, 1)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Order and Degree Definition|Implicit Differentiation Implicit Domain|Cross Multiplication Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Implicit Domain' and 'Cross Multiplication Trap' in finding the order and degree of a differential equation.

Students often cross-multiply to clear denominators and form a polynomial in derivatives, arriving at $0 = (y'' - y')((y')^3 - x)$ .

A common mistake is to stop here and identify the highest derivative as $y''$ with a power of 1, incorrectly concluding the order is 2 and degree is 1.

However, the original equation's denominator dictates the implicit domain constraint $y'' - y' \neq 0$ .

Recognizing this forces the student to divide out the extraneous factor, dropping the order to 1 and revealing the true degree of 3.

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