- #1
phosgene
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Homework Statement
I have to use Picard's method to find an approximate solution to
[itex]\frac{dy}{dx}=2xy[/itex]
and then compare it to the exact solution, given that [itex]y(0)=2[/itex]
Homework Equations
Picard's method:
[itex]y_{n}=y_{0} + ∫^{x}_{x_{0}} f(\phi,y_{n-1}(\phi))d\phi[/itex]
The Attempt at a Solution
So I go over Picard's method about 3 times and get the result:
[itex]2(1+x^2 + \frac{x^4}{2} + \frac{x^6}{6})[/itex]
which looks like it miiiight be going towards:
[itex]2(\sum ^{\infty}_{n=0} \frac{x^{2n}}{n!}) = 2e^{x^2}[/itex]I then solve the DE exactly by noting that the DE is separable, so I separate and integrate to obtain:
[itex]y= e^{x^2}+C[/itex]
Using my initial condition, I find that C=1. So, my question is, have I done this correctly? I have a feeling that I'm supposed to get the same answer for both.