Consider the differential equation dy dt =(y-2)(y+1)(t-1)

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the differential equation dydt=(y−2)(y+1)(t−1).\frac{dy}{dt}=(y-2)(y+1)(t-1). Information about the slope field shows that the factor (t−1)(t-1) changes sign at t=1t=1. (a) Determine all constant (equilibrium) solutions of the differential equation. (b) Explain why the stability of each equilibrium solution depends on whether t<1t<1 or t>1t>1. In your explanation, reference the sign of (t−1)(t-1) and its effect on dydt\dfrac{dy}{dt} near each equilibrium value. (c) Let y(0)=3y(0)=3. Describe qualitatively how the solution behaves for 0≤t<10\le t<1 and for t>1t>1. Include comments on any intervals where the solution is increasing or decreasing and its limiting behavior as t→∞t\to\infty. Support your description with reasoning based on parts (a) and (b). (d) Let y(0)=0y(0)=0. Repeat the analysis of part (c) for this initial condition.

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