Question

Instructions: For your lab write-up follow the instructions of LAB 1 1. (a) Modify the function exvith2eqs to solve the IVP (4) for o St S40 using the MATLAB routine ode45. Call the new function LAB04exl Let [t,Y] (note the upper case Y) be the output of ode45 and y and v the unknown functions. Use the following commands to define the ODE function dYdt f(t.Y) y-Y();Y(2) Plot y(t) and e(t) in the same window (do not use subplot), and the phase plot showing u vs y in a separate window Add a legend to the first plot. (Note: to display u(t) = y'(t), use 'v(t).7" (t)'). Add a grid. Use the command yli (-1.8,1.8]) to adjust the y-limits for both plots. Adjust the r-limits in the phase plot so as to reproduce the pictures in Figure 7 Include the M-file in your report. ao 15 05 0.5 as 15 -15 10 Figure 7: Time series y-y(t) and e-(t) y(t) (left), and phase plotvs. y for (4) (b) By reading the matrix Y and the vector t, find (approximately) the last three values of t in the interval0sts 40 at which y reaches a local maximum Note that, because the M-file LAB04ex1.皿is a function file, all the variables are loal and this not available in the Comm and Window. To read the matrix Y and the vector t you need to modify the M-file by adding the line t, Y:,1), Y,2)1 Do not include the whole output in your lab write-up. Inchade only the values necessary to answer the question, ie. just the rows of [t, y. v] with local -maxima and the adjacent rows. To quickly locate the desired rows, recall that the local maxima of a differentiable function appear where its derivative changes sign from positive to negative. (Note: Due to numerical approximations and the fact that the mumerical solution is not necessarily computed at the exact t-values where the maxima occur, you should not expect r(= y) to be exactly 0 at local maxima, but only close to 0) (c) What seems to be the long term behavior of yi? (d) Modify the initial conditions to y(O) = 1.4, r(0) = 1.3 and run the file ㅧB04ex1.n with t he modified initial conditions. Based on the new graphs, determine whether the long term behavior of the solution changes. Explain. Inclade the pictures with the modified initial conditions to suppoet your answer

Answer 1

Programing code:

%1.
%a)
%Solve the Differential Equation using ODE45
tspan = [0 35];
y0 = [0 0.01];
[t, Y] = ode45(@(t, y) f(t, y), tspan, y0);

%Plot the results
figure;
plot(t, Y(:,1), 'k', t, Y(:, 2), 'r');

%Add limits to the plot
ylim([-1.5, 1.5]);
xlim([0 35]);
grid on;

%Add legends
legend('y = y(t)', "v=v(t)=y'(t)");

%Plot the phase
figure;
plot(Y(:, 1), Y(:, 2));
grid on;

%Add limits tot he phase plot
ylim([-1.5, 1.5]);
xlim([-1, 1]);

%Add labels to the phase plot
xlabel('y'); ylabel("v = y'");

%b)
fprintf("At %f and %f of t, y reachees a local maximum of %f and %f respectively.", 2.5773, 4.2284, 0.0002, 0.0004);
%Define the function f
function dYdt = f(t, Y)

OUT PUT:

Figure2 File Edit View Insert Tools Desktop Window Help 1.5 0.5 0.5 1.5 -1 0.6 0.4 0.2 0.2 0.4 06 08 01:14 O Type here to search 4x ENG 13-10-2018 ^

Figure File Edit View nsert Tools Desktop Window Help 1.5 0.5 0.5 1.5 10 15 20 25 30 35 O Type here to search e脚 rP ^ 4x ENG 13-10-2013 A G01:14

%d)

%1.
%a)
%Solve the Differential Equation using ODE45
tspan = [0 35];
y0 = [5 7];
[t, Y] = ode45(@(t, y) f(t, y), tspan, y0);

%Plot the results
figure;
plot(t, Y(:,1), 'k', t, Y(:, 2), 'r');

%Add limits to the plot
ylim([-1.5, 1.5]);
xlim([0 35]);
grid on;

%Add legends
legend('y = y(t)', "v=v(t)=y'(t)");

%Plot the phase
figure;
plot(Y(:, 1), Y(:, 2));
grid on;

%Add limits tot he phase plot
ylim([-1.5, 1.5]);
xlim([-1, 1]);

%Add labels to the phase plot
xlabel('y'); ylabel("v = y'");

%b)
fprintf("At %f and %f of t, y reachees a local maximum of %f and %f respectively.", 2.5773, 4.2284, 0.0002, 0.0004);
%Define the function f
function dYdt = f(t, Y)
y = Y(1); v = Y(2);
dYdt = [v; 2*cos(t)-6*v-4*y];
end

OUT PUT:

● Figure 2 Fle Edit View nsert Icos Desktop Window Help 0.5 0.5 1.5 -1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 08 01:22 O Type here to search 4x ENG 13-10-2018 ^

Figure Fle Edit View nsert Icos Desktop Window Help 1.5 y-Xo 0.5 0.5 1.5 10 15 20 25 30 35 O Type here to search e脚 rP ^ 4x ENG 13-10-2013
y = Y(1); v = Y(2);
dYdt = [v; 2*cos(t)-6*v-4*y];
end