Matlab Assignment 1

• Follow the instructions in Using Matlab for first order ODEs.
• You need to download m-files dirfield.m and Euler1.m and put them in the same directory as your m-files. First try out the examples below. Then modify these examples for the homework problems.
  Instructions for using dirfield
  Example for using Euler1
• Matlab needs to know where to look for m-files:
  • Either use the current folder field in the desktop toolbar (click on the "..." to the right of the field), see Matlab desktop window with current folder field
  • or use the addpath command, see How to tell Matlab where to look for m-files
• You must prepare the solution for each problem as an m-file, e.g., prob2.m.
• You can then use publish('prob2a') to generate an html file containing commands, output and graphics. You can open this html file in your web browser and print it from your web browser.

Hints

• Problem 1: Use y(0)=-1/2 as initial condition for (d), (e).
• Problem 2(b): To solve the initial value problem from t=0 to t=-5 use ode45(f,[0,-5],y0).
  
• Problem 3:
  3(a): First find the function &Psi(t,y) by hand and use this for (b), (c), (d). Then try to use dsolve. If you have an older version of Matlab you should get an answer which gives the function &Psi(t,y) in a strange format. If you have a newer version of Matlab, dsolve will return an empty solution and the warning "Explicit solution could not be found.".

  Example problem: exact ODE 2y·y' + 2t = 0, initial condition y(3) = -4
  You can check by hand that this ODE is indeed exact and then find Ψ(t,y) = t² + y². Then the general solution in implicit form is t² + y² = C, C arbitrary.

  • Plot graphs of the general solution for various values of C:

       syms t y             
       psi = t^2 + y^2    
       ezcontour(psi,[-5 5 -5 5])      % plot contours of psi for t in [-5,5], y in [-5,5]  
       hold on
       axis equal                      % use same size units on both axes
    

  • Find C for given initial condition y(3)=-4:

       C = subs(psi,{t,y},{3,-4})      % substitute t=3, y=-4 in psi
    

  • Find y(2) using fzero:

       F = subs(psi-C,t,2)             % substitute t=2 in psi-C
       Fi = inline(vectorize(F))       % turn symbolic expression F into inline function Fi (needed for fzero)
       y1 = fzero(Fi,-5)               % find y1 such that F=0, near y=-5 
       plot(2,y1,'o')                  % mark point (2,y(2)) with circle
       hold off