Question

ODE

From van der pol’s equation

4. For μ = 1 .5 and the initial conditions x(0) = 0.a and x (0)= 0b , where a and b are before the last and the last digits of your student ID (replace zeros by 9), respectively, use the Euler's method to convince yourself that the trajectory is "attracted" to the closed orbit from question 3 for μ = 1 .5 from inside. Supply the table of the first 20 values of x(t) and x(t) of your numerical solution. Sketch on the same phase-plane both trajectories: the trajectory you obtain ed with the initial condition x()-0.a and x(o)-0b and the closed trajectory with the initial conditions(0)-2, x"(0)-0.

Answer 1

%%Matlab code for Euler's forward clear all close all mu for Vander pol equation mu-1.5; %function for Euler equation solution fl-i (x,y,t) yi f2-4 (x,y,t) mu*(1-x*2)*y-x; all step size 加0.01; %Initial values change as per student ID of the for 0.a and 0.b where a and b are before the last and last digit of Student ID :0 0.3; %0.a as per student id yo=0.4; %0.b as per student id %initial t t0-0 t end values tend-20; tn-to:h:tend; % Euler steps x1_result(1)-x0; x2 result(1)=y0; t_result (1)-t0; %loop for Euler step for finding the solution for i-1:length(tn)-1 t-result(i+1)= t-result(i)th; x1_result(i +1 )EX1-result (i) +h *double ( f1 (x1-result (i) , x2-result( į), t-result( İ)) ) ; x2 result(i +1)-x2_result (i)th double (£2 (x1 result (i), x2_result (i),t_result(i))); end %printing the result for solving Vander pol equation fprintfprinting the result for solving Vander pol equation for x0-0.3 dx (0)/dt-0.4 n' fprintf (ttimeltx (t) \tdx (t)/dt\n') for i-l:100:length (x1_result) In',t_result(i), xl_result(i),x2_result (i)) end Plotting exact solution and numerical solution figure(1) plot (x1 result,x2 result) xlabel(x) ylabel( dx/dt') title( 'phase potrait plot for x(0)#0.3 dx(0) /dt-0.4.) tall step size h-0.01:

phase potrait plot for x0:0.3 dpja:0.4 2.5 2 1.505 051525 phase potrait plot for x(0)-2 dx(0)/dt 0 2.5 21.5 50515 2 25

%%Matlab code for Euler's forward
clear all
close all
%mu for Vander pol equation
mu=1.5;
%function for Euler equation solution
f1=@(x,y,t) y;
f2=@(x,y,t) mu*(1-x^2)*y-x;
%all step size
h=0.01;
%Initial values change as per student ID of the for 0.a and 0.b
%where a and b are before the last and last digit of Student ID
x0=0.3; %0.a as per student id
y0=0.4; %0.b as per student id
%initial t
t0=0;
%t end values
tend=20;
tn=t0:h:tend;

% Euler steps
x1_result(1)=x0;
x2_result(1)=y0;
t_result(1)=t0;

%loop for Euler step for finding the solution
for i=1:length(tn)-1
t_result(i+1)= t_result(i)+h;
x1_result(i+1)=x1_result(i)+h*double(f1(x1_result(i),x2_result(i),t_result(i)));
x2_result(i+1)=x2_result(i)+h*double(f2(x1_result(i),x2_result(i),t_result(i)));
end

%printing the result for solving Vander pol equation
fprintf('printing the result for solving Vander pol equation for x0=0.3 dx(0)/dt=0.4\n')
fprintf('\ttime\tx(t)\tdx(t)/dt\n')
for i=1:100:length(x1_result)
    fprintf('\t%.2f\t%.4f\t%0.4f\n',t_result(i),x1_result(i),x2_result(i))
end

%Plotting exact solution and numerical solution
figure(1)
plot(x1_result,x2_result)
xlabel('x')
ylabel('dx/dt')
title('phase potrait plot for x(0)=0.3 dx(0)/dt=0.4')

%all step size
h=0.01;
%Phase potrait plot for initial condition x0=0 and y0=0
x0=2;
y0=0;
%initial t
t0=0;
%t end values
tend=20;
tn=t0:h:tend;

% Euler steps
x1_result(1)=x0;
x2_result(1)=y0;
t_result(1)=t0;

%loop for Euler step for finding the solution
for i=1:length(tn)-1
t_result(i+1)= t_result(i)+h;
x1_result(i+1)=x1_result(i)+h*double(f1(x1_result(i),x2_result(i),t_result(i)));
x2_result(i+1)=x2_result(i)+h*double(f2(x1_result(i),x2_result(i),t_result(i)));
end

%printing the result for solving Vander pol equation
fprintf('\nprinting the result for solving Vander pol equation for x0=2 dx(0)/dt=0\n')
fprintf('\ttime\tx(t)\tdx(t)/dt\n')
for i=1:100:length(x1_result)
    fprintf('\t%.2f\t%.4f\t%0.4f\n',t_result(i),x1_result(i),x2_result(i))
end

%Plotting exact solution and numerical solution
figure(2)
plot(x1_result,x2_result)
xlabel('x')
ylabel('dx/dt')
title('phase potrait plot for x(0)=2 dx(0)/dt=0')

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