A population model is governed by the logistic differential equation dy dx =y(1-y)

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

Problem

A population model is governed by the logistic differential equation dydx=y(1−y),y(0)=0.20.\frac{dy}{dx}=y(1-y),\qquad y(0)=0.20. Euler’s method with step size h=0.10h=0.10 will be employed to approximate the solution for 0≤x≤1.00\le x\le1.0. (a) Show that d2ydx2=y(1−y)(1−2y)\dfrac{d^2y}{dx^2}=y(1-y)(1-2y). Express your result solely in terms of yy. (b) Identify the value(s) of yy at which the solution curve changes concavity. Using those values and the initial condition, describe on which portions of the interval 0≤x≤1.00\le x\le1.0 the solution is concave up and on which portions it is concave down. (c) Using Euler’s method with h=0.10h=0.10, calculate approximate values for yy at x=0.10,0.20,…,1.00x=0.10,0.20,\dots,1.00. Organize your results in a table. (d) Determine the first Euler step at which the local truncation error switches from negative to positive. Justify your answer by combining the numerical results from part (c) with the concavity analysis from part (b).

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