Before You Begin

Question #68621

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be a real-valued function satisfying the differential equation $x \frac{dy}{dx} = y + \frac{x^2}{y} f(x)$ for $x \in \mathbb{R} \setminus \{0\}$ , where $f(x)$ is a continuous odd function such that $\lim_{x \to 0} \frac{f(x)}{x}$ exists. The solution is given by $y^2 = x^2 \left( 2 \int_0^x \frac{f(t)}{t} dt + C(x) \right)$ , where the integration parameter $C(x)$ is defined as $C(x) = \lambda [x]^2 + \mu [x]$ , with $[ \cdot ]$ denoting the greatest integer function, and $\lambda, \mu \in \mathbb{R}$ . Let $I_n = \int_{-n}^{n} \frac{y^2}{x^2} dx$ . If $I_2 = 2$ and $I_3 = 13$ , find the value of $\int_0^{\lambda + \mu} (3t^2 - 2t) dt$ .

Options

A.
18
B.
24
C.
12
D.
36

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Integration Constant Trap|Homogeneous Function Homogeneity|Greatest Integer Function (GIF)|Even and Odd Function Symmetry

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the student's ability to handle piecewise constants of integration in differential equations, a trap where students often assume the integration constant must be a global scalar.

By defining C(x) using the Greatest Integer Function (GIF), the solution becomes piecewise.

The problem also integrates the concept of homogeneous ODE substitution (y=vx) and leverages even/odd function symmetry.

The core trap is recognizing that the integral of the unknown function f(x) vanishes over symmetric intervals because f(x)/x is even, making its antiderivative odd.

Failing to identify this symmetry or incorrectly evaluating the GIF integrals over negative domains (e.g., incorrectly assuming [-x]^2 = [x]^2) will lead to step-wise failure in finding the parameters.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Integration Constant Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free