Solve The Differential Equation. Dy Dx 6x2y2 Apr 2026

This solution is perfectly fine for small (x). But as (x) approaches ( \sqrt[3]{\frac12} ) from below, the denominator (1 - 2x^3 \to 0^+), so (y \to +\infty).

At first glance, the differential equation [ \frac{dy}{dx} = 6x^2 y^2 ] might look like a simple textbook exercise. And it is. But hidden within its simplicity is a beautiful tension—one that touches on growth, separation, and a surprising warning about the limits of prediction. Step 1: Separation of Variables The equation is separable , meaning we can rearrange it so that all (y) terms are on one side and all (x) terms on the other. [ \frac{dy}{y^2} = 6x^2 , dx ] solve the differential equation. dy dx 6x2y2

That is, . At (x = \left(\frac12\right)^{1/3} \approx 0.7937), the population (or whatever (y) represents) blows up. This solution is perfectly fine for small (x)