Question 4
Part (a)
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):
x^{'}=dx/dt=-ky
y^{'}=dy/dt=kx
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
Eigenvalues_Q4a =
-k*1i
k*1i
Eigenvectors_Q4a =
[ -1i, 1i]
[ 1, 1]
DiagMat =
[ -k*1i, 0]
[ 0, k*1i]
Q4b: Solving the system of ODE
syms x(t) y(t)
%defining the odes
ode1=diff(x,t)==-4*y;
ode2=diff(y,t)==4*x;
odee=[ode1,ode2];
%defining the initial conditions
cond1=x(0)==1;
cond2=y(0)==0;
cond=[cond1,cond2];
%solving the ODEs using dsolve()
soln=dsolve(odee,cond);
x=soln.x;
y=soln.y;
% plotting x(t) and y(t)
t1=0:0.001:5;
figure(1)
plot(t1,subs(x,t,t1))
xlabel("t")
ylabel("x(t)")
title("Plot of x(t) against time, t")
figure(2)
plot(t1,subs(y,t,t1))
xlabel("t")
ylabel("y(t)")
title("Plot of y(t) against time, t")
Question 1
Part a:
Let: (Shiwnarain, 2017)
x_{1} (t)be the number of larvae at time t
x_{2} (t) be the number of polliwogs at time t
x_{3} (t) be the number of catigorgons at time t
x_{4} (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
% x1---larvae
% x2---polliwogs
% x3---catigorgons
% x4---demidogs
eqn1=diff(x1,t)==-1/30*x1+5/2*x4;
eqn2=diff(x2,t)==1/30*x1-1/3*x2;
eqn3=diff(x3,t)==1/3*x2-1/2*x3;
eqn4=diff(x4,t)==1/2*x3-1/365*x4;
odee1=[eqn1,eqn2,eqn3,eqn4];
%defining the initial conditions
condi1=x1(0)==20;
condi2=x2(0)==5;
condi3=x3(0)==1;
condi4=x4(0)==0;
condi=[condi1,condi2,condi3,condi4];
% solving the ODEs
soln1=dsolve(odee1,condi);
% obtaining the values of x1, x2, x3 and x4 from the solution
x1=vpa(soln1.x1,3);
x2=vpa(soln1.x2,3);
x3=vpa(soln1.x3,3);
x4=vpa(soln1.x4,3);
% plotting x4---demidogs against time
t2=0:1:400;
figure (3)
plot(t2,subs(x4,t,t2))
xlabel("t")
ylabel("x4(t)")
title("Plot of demidogs (x4) population against time, t")
disp("The number of demidogs after 7 days")
Demidogs_7=uint64(abs(subs(x1,t,7)))
disp("The number of demidogs after 30 days")
Demidogs_30=uint64(abs(subs(x1,t,30)))
disp("The number of demidogs after 365 days")
Demidogs_365=uint64(abs(subs(x1,t,365)))
Warning: Imaginary parts of complex X and/or Y arguments ignored
The number of demidogs after 7 days
Demidogs_7 =
uint64
70
The number of demidogs after 30 days
Demidogs_30 =
uint64
4970
The number of demidogs after 365 days
Demidogs_365 =
uint64
18446744073709551615
Part b
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];
Eigenvalues_Q1b=eig(A)
Eigenvalues_Q1b =
-0.6273 + 0.0000i
-0.2118 + 0.2775i
-0.2118 - 0.2775i
0.1815 + 0.0000i
References
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 services.
The experienced team of AssignmentHippo has got your back 24*7. Get connected with our Live Chat support executives to receive instant solutions for your assignment problems.
We can build quality assignments in the subjects you're passionate about. Be it Programming, Engineering, Accounting, Finance and Literature or Law and Marketing we have an expert writer for all.
Get premium service at a pocket-friendly rate. At AssignmentHippo, we understand the tight budget of students and thus offer our services at highly affordable prices.