For the differential equation

MATHEMATICS · Single correct · IB Diploma Programme · Mathematics: Applications and Interpretation · Calculus · Calculus

Problem

For the differential equation dydx=y2−4\dfrac{dy}{dx}=y^{2}-4, let y(0)=1y(0)=1. Which of the following long-term behaviors is most consistent with the slope field and the given initial condition? (a) The solution decreases without bound, so y→−∞y\to-\infty as x→∞x\to\infty. (b) The solution decreases and approaches y=−2y=-2 as x→∞x\to\infty, without crossing that value. (c) The solution decreases and approaches y=0y=0 as x→∞x\to\infty. (d) The solution eventually increases without bound, so y→∞y\to\infty as x→∞x\to\infty.

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