The differential equation ( dy dx =y^ 2 -4 ) defines a family of solution curves.

MATHEMATICS · Multi-part · IB Diploma Programme · Mathematics: Applications and Interpretation · Calculus · Calculus

Problem

The differential equation dydx=y2−4\displaystyle \frac{dy}{dx}=y^{2}-4 defines a family of solution curves. (1) Identify all equilibrium solutions and verify that they satisfy the differential equation. (2) For each equilibrium solution found in part (a), determine whether it is attracting or repelling. Justify your answer by referring to the sign of dydx\dfrac{dy}{dx} in regions immediately above and below the equilibrium value. (3) On a set of axes covering −2≤x≤2-2\le x\le 2 and −4≤y≤4-4\le y\le 4, sketch the slope field for the differential equation. Then draw solution curves passing through the points (0,0)(0,0) and (0,3)(0,3). Label each curve clearly. (4) Using only information from your slope field and sketches, determine lim⁡x→∞y(x)\displaystyle \lim_{x\to\infty}y(x) for the solution curve that passes through (0,0)(0,0). Provide a justification based on the behavior of the slopes and the equilibrium solutions.

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