The information given implies that:
dx/dt∝-y,x decreases at a rate proportional to y
dy/dt∝x,y increases at a rate proportional to x
The constant of proportionality is the same,k for the two equations (Balamuralitharan, 2018): Therefore the system of differential equations are (Bird, 2017):
In matrix form:
Using MATLAB we get:
Q4a:Obtaining eigenvalues and eigenvectors
syms k v1 v2
A=[0 -k; k 0];
Eigenvalues_Q4a=eig(A) %finding eigenvalues
[Eigenvectors_Q4a,DiagMat]=eig(A) %finding eigenvectors
[ -1i, 1i]
[ 1, 1]
[ -k*1i, 0]
[ 0, k*1i]
Q4b: Solving the system of ODE
syms x(t) y(t)
%defining the odes
%defining the initial conditions
%solving the ODEs using dsolve()
% plotting x(t) and y(t)
title("Plot of x(t) against time, t")
title("Plot of y(t) against time, t")
Let: (Shiwnarain, 2017)
x1 (t)be the number of larvae at time t
x2 (t) be the number of polliwogs at time t
x3 (t) be the number of catigorgons at time t
x4 (t) be the number of demidogs at time t
MATLAB code and graphs (Magrab et al., 2010)
Q1a: System of ODEs
syms x1(t) x2(t) x3(t) x4(t)
%defining the differential equations
%defining the initial conditions
% solving the ODEs
% obtaining the values of x1, x2, x3 and x4 from the solution
% plotting x4---demidogs against time
title("Plot of demidogs (x4) population against time, t")
disp("The number of demidogs after 7 days")
disp("The number of demidogs after 30 days")
disp("The number of demidogs after 365 days")
Warning: Imaginary parts of complex X and/or Y arguments ignored
The number of demidogs after 7 days
The number of demidogs after 30 days
The number of demidogs after 365 days
The coefficient matrix from part (a) is:
MATLAB Code for eigenvalues of A:
Q1b: Eigenvalues of coefficient matrix from A
A=[-1/30 0 0 5/2; 1/30 -1/3 0 0; 0 1/3 -1/2 0; 0 0 1/2 -1/365];
-0.6273 + 0.0000i
-0.2118 + 0.2775i
-0.2118 - 0.2775i
0.1815 + 0.0000i
Balamuralitharan, S. (2018). MATLAB Programming of Nonlinear Equations of Ordinary Differential Equations and Partial Differential Equations. International Journal of Engineering & Technology, 7(4.10), 773. https://doi.org/10.14419/ijet.v7i4.10.26114
Bird, J. (2017). Higher Engineering Mathematics, 8th Ed. Routledge.
Calogero, F. (2013). A linear second-order ODE with only polynomial solutions. Journal of Differential Equations, 255(8), 2130–2135. https://doi.org/10.1016/j.jde.2013.06.007
Dwork, B. (2019). On Systems of Ordinary Differential Equations with Transcendental Parameters. Journal of Differential Equations, 156(1), 18–25. https://doi.org/10.1006/jdeq.1998.3593
King, A. C., & Otto, S. R. (2013). Differential equations : linear, nonlinear, ordinary, partial. Cambridge University Press.
Magrab, E. B., Shapour Azarm, & College, M. (2010). An engineer’s guide to MATLAB. Prentice Hall.
Shiwnarain, M. (2017). Life Cycle Of A Butterfly: Stages Of Life. Science Trends. https://doi.org/10.31988/scitrends.3973
Yamauchi, Y. (2012). Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations. International Journal of Differential Equations, 2012, 1–16. https://doi.org/10.1155/2012/417261
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