Question

Use MATLAB’s ode45 command to solve the following non linear 2nd order ODE:

y'' = −y' + sin(ty)

where the derivatives are with respect to time. The initial conditions are y(0) = 1 and y ' (0) = 0. Include your MATLAB code and correctly labelled plot (for 0 ≤ t ≤ 30). Describe the behaviour of the solution.

Under certain conditions the following system of ODEs models fluid turbulence over time:

dx / dt = σ(y − x)

dy / dt = rx − y − xz

dz / dt = xy − bz

where x, y and z model movement of fluid in three dimensions. Assume σ = 10, r = 28 and b = 8 3 and initial conditions of x(0) = 2, y(0) = 4 and z(0) = 6. Use MATLAB’s ode45 command to solve this system of nonlinear ODEs and plot your solution for 0 ≤ t ≤ 5 (include your MATLAB code and correctly labelled plot). Comment on the solution.

Answer 1

%for 2nd order ode
ode = @(t,y)[y(2);-y(2)+sin(t*y(1))];
init=[1;0];
tspan=[0,30];
[t,y]=ode45(ode,tspan,init);
plot(t,y(:,1),t,y(:,2))
legend('y(t)','y''(t)')
xlabel('t')
ylabel('y and y''')

%for fluid turbulence system
sigma=10;
r=28;
b=83;
f = @(t,xyz)[sigma*(xyz(2)-xyz(1));r*xyz(1)-xyz(2)-xyz(1)*xyz(3);
    xyz(1)*xyz(2)-b*xyz(3)];
init=[2;4;6];
tspan=[0,5];
[t,XYZ]=ode45(f,tspan,init);
figure
plot(t,XYZ(:,1),t,XYZ(:,2),t,XYZ(:,3))
legend('x','y','z')
xlabel('t')
ylabel('movement in 3 dimensions')