Use matlab Growth of biological samples can me modeled using an autonomous first

Question

Use matlab Growth of biological samples can me modeled using an autonomous first order differential equation dy/dt=y(1-y/3),y(0)=1 Obtain a solution to this DE using (a) Euler's method using At=.2 and (b) ODE with At=0.2. The analytical solution is given as y(3)=3-6/exp(t)+2 Estimate the MSE in each case. Use a time window from t=0 to t=6. lot the Euler's solutions, ODE solution and analytical solution on a single plot, annotated. The MSE values must also be displayed. Use Matlab to get the EQUILIBRIUM value and determine the time instant when it is reached. Find how long does it take the growth to double from 1.2 to 2.4? Provide answers to part (2) and (3) as comments or display them on the plot.

Explanation / Answer

1.

function sequence = eulers(f,a,b,y0,n)

h=(b-a)/n; % interval length

x(1)=a;

y(1)=y0;

for K=1:n

x(n+1)=x(n)+h;

y(n+1)=y(n)+h*f(x(n),y(n));

end

sequence = [x(n),y(n)]

end

clear;

S = [5, 10, 30, 200]; % n-values

numx = length(S);

Y = zeros(1,numx);

for T = 1:numx

Y(T) = eulers(@(x,y) 2*x*y*(1-y),0,2,2, S(T));

end

% calculate the exact solution

fdash = @(x,y) 2*x*y*(1-y);

tspan = [0,2];

yzero = 2;

[xexact,yexact] = ode45(fdash,tspan,yzero);

plot(x,y,'g',xexact,yexact, 'k')

title(['Eulers Method vs Exact Solution'])

legend('Approximate','Exact');

a=0;

b=2;

y0=2;

n=50;

f=@(x,y) 2*x*y*(1-y);

h=(b-a)/n; % interval length

x(1)=a;

y(1)=y0;

for K=1:n

x(K+1)=x(K)+h;

y(K+1)=y(K)+h*f(x(K),y(K));

end

plot (x,y);

2.

step(sys)

step(sys,Tfinal)

step(sys,t)

step(sys1,sys2,...,sysN)

step(sys1,sys2,...,sysN,Tfinal)

step(sys1,sys2,...,sysN,t)

y = step(sys,t)

[y,t] = step(sys)

[y,t] = step(sys,Tfinal)

[y,t,x] = step(sys)

[y,t,x,ysd] = step(sys)

[y,...] = step(sys,...,options)

t = (0:0.1:10)';

[y1, ~, ~, ysd1] = step(sys1,t);

[y2, ~, ~, ysd2] = step(sys2,t);

plot(t, y1, 'b', t, y1+3*ysd1, 'b:', t, y1-3*ysd1, 'b:')

hold on

plot(t, y2, 'g', t, y2+3*ysd2, 'g:', t, y2-3*ysd2, 'g:')

u0 = 1;

[X,~,r] = findop(nlsys, 'steady', 1);

y0 = r.SignalLevels.Output;

sys = linearize(nlsys,u0,X);

opt = stepDataOptions;

opt.InputOffset = u0;

opt.StepAmplitude = 0.1;

t = (0:0.1:10)';

ynl = step(nlsys, t, opt);

plot(t, ynl, t, yl+y0)

legend('Nonlinear', 'Linear with offset')

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