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Question #41418

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be a differentiable function on $\mathbb{R}$ satisfying the differential equation $(x-2) \frac{dy}{dx} - y = \frac{(x-2)^7 + 313(x-2)^5 + 21(x-2)^3}{(x-2)^4 + 4(x-2)^2 + 5} - 1$ If $y(3) = 2024$ , then the value of $y(4) + y(0)$ is

Options

A.
2
B.
4048
C.
0
D.
4

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

Partial Fractions Decomposition|Homogeneous Function Homogeneity|Geometric Interpretation|Polynomial Function Parity

Why This Question Is Tricky

Common Mistakes

This question tests the ability to recognize reducible homogeneous differential equations and exploit the geometric properties of definite integrals (parity).

The primary trap is the intimidating rational function and the initial condition $y(3) = 2024$ , which baits students into performing a tedious partial fraction decomposition to find the constant of integration.

However, by substituting $X = x-2$ and dividing by $X^2$ , the right-hand side becomes an odd function.

Integrating over the symmetric interval $[-2, 2]$ causes the integral to vanish, completely bypassing the need to evaluate the rational function or use the red herring initial condition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Partial Fractions Decomposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free