A population P, in millions, at a given time+ years years, satisfies the differential equationdpJE = P(1-P)atInitial the population is a quarter of a millich, i.l.P(0)=0.25Applying two iterations of the 4th oreler Runge-Kutta Method, with a step size h=1, to theabove problem yields the following results:Iteration 1:=КогKoa ===КозКонP(1)~P₁₂=Iteration aК11K₁1 =K₁2 =K13 =K14 =P(a)~ B₂ =-

Answered: A population P, in millions, at a given…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A population P, in millions, at a given time
+ years years, satisfies the differential equation
dp
JE = P(1-P)
at
Initial the population is a quarter of a millich, i.l.
P(0)=0.25
Applying two iterations of the 4th oreler Runge-
Kutta Method, with a step size h=1, to the
above problem yields the following results:
Iteration 1:
=
Ког
Koa =
=
=
Коз
Кон
P(1)~P₁₂=
Iteration a
К11
K₁1 =
K₁2 =
K13 =
K14 =
P(a)~ B₂ =
-

Expert Solution

Step 1

For a differential equation of the form: $y' = f (x, y)$ , the Runge-Kutta 4 method uses the following formulas to perform the iterations and to approximate the values of y at specific values of x.

$\begin{array}{rcl} k_{1} & = & h f (x_{0}, y_{0}) \\ k_{2} & = & h f (x_{0} + \frac{h}{2}, y_{0} + \frac{k_{1}}{2}) \\ k_{3} & = & h f (x_{0} + \frac{h}{2}, y_{0} + \frac{k_{2}}{2}) \\ k_{4} & = & h f (x_{0} + h, y_{0} + k_{3}) \\ y_{1} & = & y_{0} + \frac{1}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4}) \end{array}$