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Question #41153

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be a continuous function defined on $[1, \infty)$ which satisfies the differential equation $x \frac{dy}{dx} - y = x^3 [x]$ for all non-integer $x$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . If $y(1) = 0$ , then the value of $y(6)$ is

Options

A.
375
B.
430
C.
645
D.
105

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

Greatest Integer Function Traps|Substitution Logic in Homogeneous DE|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Integration Constant Trap' within a piecewise Homogeneous Differential Equation.

Students commonly make the mistake of integrating $x[x]$ as $[x]x^2/2 + C$ and assuming a single global constant of integration $C$ works for the entire domain (leading to the trap answer 645).

However, because the Greatest Integer Function $[x]$ is piecewise constant, the DE must be solved independently on each unit interval $[n, n+1)$ with distinct constants $C_n$ .

The requirement of continuity at the integer boundaries forces these constants to link together, creating a recurrence relation ( $C_n - C_{n-1} = -n^2/2$ ).

This effectively penalizes rote application of integration formulas and rewards rigorous interval-based calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Greatest Integer Function Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free