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Question #41525

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the solution to the differential equation $x \frac{dy}{dx} - y = x \tan\left(\frac{\pi}{4} + \frac{y}{2x}\right), \quad x < 0$ with the initial condition $y(-1) = -\pi$ . Find the value of $y\left(-\frac{1}{2}\right)$ .

Options

A.
$-\frac{7\pi}{12}$
B.
$\frac{\pi}{12}$
C.
$-\frac{11\pi}{12}$
D.
$-\frac{5\pi}{12}$

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

Logarithmic Integration via Trig Manipulation|Homogeneous Function Homogeneity|General vs Particular Solutions

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate logarithmic integration and trigonometric manipulation within a homogeneous differential equation.

The primary trap lies in the transition from the general to the particular solution.

After finding $\sin v = -1/2$ , students frequently default to the principal value $v = -\pi/6$ or another incorrect branch like $11\pi/6$ .

However, by analyzing the initial condition $v(-1) = \pi$ and the derivative's sign, one must deduce that $v(x)$ increases continuously into the third quadrant, making $7\pi/6$ the only valid branch.

It also tests the careful handling of absolute values in logarithmic integration for $x < 0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Logarithmic Integration via Trig Manipulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free