adjuster jack:
2 - Setting to Disable prevents pasting anything in this space. That's no good.
Of course, it is your choie whether you have it set on enable, disable, or prompt. I did, however, for grins and giggles go and disable, and I was still able to paste the below from a page on the Picard iterative process for solving first-case differential equations. (Now I am going to go back and reset to "prompt" since that's my preferred setting.)
Pasted with setting disabled:
Indeed, often it is very hard to solve differential equations, but we do have a numerical process that can approximate the solution. This process is known as the Picard iterative process.
First, consider the IVP
It is not hard to see that the solution to this problem is also given as a solution to (called the integral associated equation)
The Picard iterative process consists of constructing a sequence of functions which will get closer and closer to the desired solution. This is how the process works:
- (1)
- for every x;
- (2)
- then the recurrent formula holds
for .
Example: Find the approximated sequence , for the IVP
.
Solution: First let us write the associated integral equation
Set . Then for any , we have the recurrent formula
We have , and
We leave it to the reader to show that
We recognize the Taylor polynomials of (which also get closer and closer to) the function
