Guys I really need help with understanding how to use this method for my O.D.E. class. The main part I don't understand is, I don't get what f(t,Yn(t)) is supposed to be. I also do not understand how to do the next iteration after the first and the one after that and so on. I know their is a pattern but could someone show me how to do all this stuff to help me prepare for a Test? For instance
dy/dx = (x^2)(y)
y(0) = 1
This is from my notes, Why is f(t,Yn(t)) = to [t^2 - 1]? What procedure is there for figuring this out? Also if possible, could you tell it to me in a way for which it will work for any Picard Iteration problem the professor gives us? I just need to know how f(t,Yn(t)) becomes what it is.
Here's a procedure that should always work. Start with the equation . For convenience, replace the variable x by t, so that y is a function of t: . Integrate both sides from 0 to x: , or .
The idea now is to start with an approximate solution which you substitute into the integral, getting a new function which you then substitute into the integral to get another function , and so on.
The first approximation is just the constant function 1 (because this is given as the value of y at x=0). Then
. So . Next,
, and so on.
I just figured out that the example the professor showed us in class was wrong. You are right, it should most definitely not be what I have above. Thanks guys for your input~! It really helped me get this.
Originally Posted by HallsofIvy