Mathematics

Mathematics

Problem

A planar trajectory of a particle is modeled implicitly by sin⁡(xy)+x2=3.\sin(xy)+x^{2}=3. Only points with integer coordinates are relevant in the laboratory setup. Investigate the behavior of the particle at such points. (a) Determine every point with integer coordinates that lies on the curve. (b) For each point found in part (a), determine whether the point is critical. Show complete algebraic justification. (c) State the collection of critical points with integer coordinates, giving a brief reason for discarding any extraneous solutions discovered during your work.
Intuitive explanation

The key is to recognize that integer coordinates are extremely restrictive when combined with a transcendental equation. Since sin⁡(xy)\sin(xy) must equal 3−x23 - x^2, and sine is bounded between −1-1 and 11, you need −1≤3−x2≤1-1 \leq 3 - x^2 \leq 1, which severely limits possible xx values. Once you find which integers xx can take, you can determine corresponding integer yy values by solving for when sin⁡(xy)\sin(xy) yields an integer-compatible result. This constraint-based approach transforms an implicit curve into a discrete set of points.

Intuitive explanation

The key is to recognize that integer coordinates are extremely restrictive when combined with a transcendental equation. Since sin⁡(xy)\sin(xy) must equal 3−x23 - x^2, and sine is bounded between −1-1 and 11, you need −1≤3−x2≤1-1 \leq 3 - x^2 \leq 1, which severely limits possible xx values. Once you find which integers xx can take, you can determine corresponding integer yy values by solving for when sin⁡(xy)\sin(xy) yields an integer-compatible result. This constraint-based approach transforms an implicit curve into a discrete set of points.

Browse the problem library