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
Our motto is deliver assignment on Time. Our Expert writers deliver quality assignments to the students.
Get reliable and unique assignments by using our 100% plagiarism-free.
Get connected 24*7 with our Live Chat support executives to receive instant solutions for your assignment.
Get Help with all the subjects like: Programming, Accounting, Finance, Engineering, Law and Marketing.
Get premium service at a pocket-friendly rate at AssignmentHippo
I was struggling so hard to complete my marketing assignment on brand development when I decided to finally reach to the experts of this portal. They certainly deliver perfect consistency and the desired format. The content prepared by the experts of this platform was simply amazing. I definitely owe my grades to them.