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Question #70167

Difficulty: 8/10 | 🟢 Free

The Question

Find the equation of the orthogonal trajectories of the family of curves $x = k |y|^a e^{b|y|}$ , where $k$ is an arbitrary non-zero constant, and $a, b$ are positive constants.

Options

A.
$b^2 x^2 + 2b|y| - 2a \ln(a+b|y|) = C$
B.
$b^2 x^2 + 2by - 2a \ln|a+by| = C$
C.
$b^2 x^2 + 2by - 2a \text{sgn}(y) \ln(a+b|y|) = C$
D.
$b^2 x^2 + 2b|y| + 2a \ln(a+b|y|) = C$

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

Orthogonal Trajectories|Tangent/Normal Slope Interplay|Modulus Sign Dependency|The Modulus Trap

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the student's ability to handle orthogonal trajectories in the presence of a modulus function, specifically targeting the 'Modulus Trap' during integration.

Students often blindly differentiate and integrate without respecting the piecewise nature of $|y|$ , leading them to drop the modulus (Option B) or incorrectly extract the signum function as a constant (Option C).

The mathematically robust path requires recognizing that $y dy$ elegantly transforms into $u du$ under the substitution $u = |y|$ , thereby maintaining sign consistency across the $y=0$ boundary and correctly reconstructing the orthogonal family.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Orthogonal Trajectories
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free