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Question #41586

Difficulty: 8/10 | 🟢 Free

The Question

Let $T_n, U_n,$ and $V_n$ be the $n^{\text{th}}$ terms of three sequences whose sums of the first $n$ terms are given by $S_n = \frac{n}{3}(n^2 - 9n + 26)$ , $S'_n = \frac{n}{6}(2n^2 - 15n + 37)$ , and $S''_n = \frac{n}{6}(2n^2 - 9n + 13)$ respectively. Consider the differential equation: $T_n \left( \frac{d^3 y}{dx^3} \right)^2 + U_n \left( \frac{d^2 y}{dx^2} \right)^4 + V_n \left( \frac{dy}{dx} \right)^5 + y = 0$ Let $O_n$ and $D_n$ denote the order and degree of this differential equation for a given $n \in \mathbb{N}$ . Find the value of $\sum_{n=1}^{5} (O_n \times D_n)$ .

Options

A.
27
B.
30
C.
31
D.
34

Check Your Answer

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|Method of Differences|Quadratic Roots in Sequences

Why This Question Is Tricky

Common Mistakes

This question tests the precise definition of the order and degree of a differential equation by linking its coefficients to sequences generated via the Method of Differences.

The primary trap lies in the 'constant collapse' of the leading coefficients.

Students often blindly assume the highest visible derivative dictates the order and degree for all $n$ , yielding an incorrect sum of 30.

A secondary trap occurs at $n=3$ , where both the third and second derivative coefficients vanish simultaneously; failing to check the discriminant roots for $U_n$ leads to an incorrect product of 8 instead of 5, resulting in the trap answer 34.

Additionally, option 27 catches students who mistakenly add the order and degree instead of multiplying them.

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