MATLAB Programs


You should search online for the basic of MATLAB if you haven't heard it before.
  • Just copy the program below and paste it in MATLAB command window.
  • You only have to change the red colored text to finish the related part of the homework.
  • You can copy line by line or the whole text. Add comments by beginning with %(All comments are green in the following code to help you understand it).
  • Blue colored text are optional parameters, you can change it to see the effects.

    • Direction Fields

    The following code is for probelm 22, in section 2.1.


    t=linspace(-2,10,10);   % Define the t region. Here 10 means we plot the direction fields for ten points.
    y=linspace(-4,4,10);    % Define the y region
    [T,Y]=meshgrid(t,y);   % Get the grid matrices
    U=ones(size(T));      % Find the horizontal component of the direction fields
    V=Y.^2-T;               % Find the vertical component of the direction fields
    % Put a dot for any multiplication, division or power, because you want to perform 'elementwise calculation' instead of 'matrxi calculation'.
    L=sqrt(U.^2+V.^2);U=U./L;V=V./L;    % Normalize the direction fields to be the same length
    quiver(t,y,U,V,0.5);   % plot the direction field, 0.5 mean the vector are rescaled to half of the original size

    Numerical solver.

    We use Problem 27, Section 2.2 as an example(y'-3y=sin t, y(0)=-3).


    [t1,y1]=ode23(@(t,y)(3*y+sin(t)),[0 pi/4],-3);       % Solve the solution on the interval [0, pi/4]
    [t2,y2]=ode23(@(t,y)(3*y+sin(t)),[0 -6*pi], -3);    % Solver the solution on the interval [-6*pi, 0]
    t=[t2(length(t2):-1:1); t1];        % Merge the indenpendent variable t
    y=[y2(length(y2):-1:1); y1];        % Mearge the dependent variable y
    plot(t,y)         % Plot the solution

    Remarks:

  • Put the equation in to normal form first.
  • Find the solution on the positive interval ([0 pi/4]) first, and then on the negative interval([0 -6*pi]), because I meage the solution with this in mind.
  • Put brackets for the definition of the function (sin(t) in above example instead of (sin t) if possible.
  • Rerun ALL the subsequent codes again to update the output, if you make the change somewhere.