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Eulers Method to Approximate Differential Eq.

  1. Mar 5, 2007 #1
    (** underscores refer to subscripts)

    Consider inital value problem y'=2xy, y(0) = 1
    Use Eulers Methos to estimate y(1) with 5 and 10 steps.

    for 5 steps, increments are by 0.2
    for 10 steps, increments are by 0.1

    y_n+1 = y_n + h f(t_n , y_n)

    where h is step size

    the answer to the DE itself is y(x) = e^(x^2)
    so these estimates should be close to e or 2.7182818...

    i understand how to go from steps to step but i do not understand how to get the values for f(t_n , y_n)
    all the examples ive looked at only had one variable so it is hard to compare

    so for

    y_1 = 1 + 0.2 f(t_1 , y_1)
    y_2 = y_1 + 0.2 f(t_2 , y_2)
    .
    .

    any help is appreciated
     
  2. jcsd
  3. Mar 5, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    One thing that may be causing trouble is that you have your algorithm in terms of "t" and "y" but your differential equation was given in terms of "x" and "y"! Either rewrite the equation as y'= 2ty or rewrite your algorithm in terms of x and y.
    In any case, you are told that f(x,y)= 2xy or that f(t,y)= 2ty.

    = 1+ 0.2 (2(0)(1))= 1

    = 1+ 0.2(2(.2)(1))= 1.08

    y_3= 1.08+ 0.2(2(.4)(1.08)= 1.08+ .1728= 1.2528

    y_4= 1.2528+ 0.2(2(.6)(1.2528))= 1.5535 ...
     
  4. Mar 5, 2007 #3
    If you are using matlab I have some code which can get you started.
     
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