Question

Let M be the integer corresponding to the first letter of your last name. For example, if your last name begins with "A", M=1 and if your last name begins with "Z", M=26.

Let k=1/M and consider the logistic equation dy/dt = k y (M - y).

Construct a single figure including

1. Title "Logistic Equation Slope Field and Euler's Method solutions by FirstName LastName" with your actual first and last names, along the top
2. Labels for the axes
3. a slope field for -3 ≤ t ≤ 3, -M/2 ≤ y ≤ 3M/2
4. Euler's method solutions for initial conditions   y(0)=5M/4, y(0)=M, y(0)=M/2, y(0)=0 and y(0)=-M/4, in each case for -3 ≤ t ≤ 3, h=0.1
5. For each initial condition, a label "y(0) = (value)" but with the actual value, at an appropriate location
6. For each solution curve, a label "y(3) ≈ (Euler's method value)" but with the actual value, or "y(3) = ?", at an appropriate location

Print on 8.5x11 paper in landscape format.

First letter of my last name is P.

Answer 1

%%Matlab code for ode solution and direction field
clear all
close all
%First letter of my last name is P
M=16;
k=1/M;

%function to be solved

f=@(t,y) (1/16).*y.*(16-y);
t_ini=-3;
t_end=3;

%Plotting Direction field
figure(1)
dirrfield(f,-3:0.2:3,-M/2:1:3*M/2)
hold on
%all initial condition
y0=[5*M/4 M M/2 0 -M/4];
%all time steps
tinit=0;tend=3; h=0.1;
%plot for different initial velocity
for i=1:length(y0)
  
   [ts,ys]=Euler(f,tinit,y0(i),tend,h);
   plot(ts,ys,'b','Linewidth',2)
   fprintf('\tUsing Euler methid For y(0)=%f , y(%2.2f) is %f\n',y0(i),ts(end),ys(end))

end

hold off
title('phase potrait plot')
xlabel('time in sec.')
ylabel('Concentration ')

%%Matlab function for direction filed
function dirrfield(f,tval,yval)
% dirfield(f, t1:dt:t2, y1:dy:y2)
%
%   plot direction field for first order ODE y' = f(t,y)
%   using t-values from t1 to t2 with spacing of dt
%   using y-values from y1 to t2 with spacing of dy
%
%   f is an @ function, or an inline function,
%     or the name of an m-file with quotes.
%
% Example: y' = -y^2 + t
%   Show direction field for t in [-1,3], y in [-2,2], use
%   spacing of .2 for both t and y:
%
%   f = @(t,y) -y^2+t
%   dirfield(f, -1:.2:3, -2:.2:2)

[tm,ym]=meshgrid(tval,yval);
dt = tval(2) - tval(1);
dy = yval(2) - yval(1);
fv = vectorize(f);
if isa(f,'function_handle')
fv = eval(fv);
end
yp=feval(fv,tm,ym);
s = 1./max(1/dt,abs(yp)./dy)*0.75;
h = ishold;
quiver(tval,yval,s,s.*yp,0,'r'); hold on;
%quiver(tval,yval,-s,-s.*yp,0,'r');
if h
hold on
else
hold off
end
axis([tval(1)-dt/2,tval(end)+dt/2,yval(1)-dy/2,yval(end)+dy/2])
end

function [t_euler,y_euler]=Euler(f,tinit,yinit,tend,h)
    %Euler method
    % h amount of intervals
    t=tinit;         % initial t
    y=yinit;         % initial y
    t_eval=tend;     % at what point we have to evaluate
    n=(t_eval-t)/h; % Number of steps
    t_euler(1)=t;
    y_euler(1)=y;
    for i=1:n
        %Eular Steps
        m=double(f(t,y));
        t=t+h;
        y=y+h*m;
        t_euler(i+1)=t;
        y_euler(i+1)=y;
    end
  
end

%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%