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How can I create a differnce equation from a differential equation? The step size h is not given and should stay a variable. For example y' = y^2 + x^2 with the known value $y(1) = 2$

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    We are using the Euler method.2012-02-08

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Here's a method.

y'(1) = 4 + 1^2

y'(1) = 5

y'' = 2 y y' + 2x

y''(1) = 2 ·2· 5 + 2 = 22

y''' = 2y'^2+2y y''+2

y'''(1) = 2 ·25 +2 ·2 ·22 +2 = 130

y^{(IV)} = 4y' y''+2y y'' + 2y y'''

$y^{(IV)}(1) = 1828$

You can go on like this. With those values we have a short Taylor Polynomial for y:

$ 2 + 5(x-1) + 11(x-1)^2 + \frac{65}{3}(x-1)^3+ \frac{457}{3} (x-1)^4$

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    Specifically, $p_0(x,y)=y$, and $p_{n+1}(x,y) = D_xp_n(x,y) + D_yp_n(x,y)(x^2+y^2)$. Oh, and I forgot a factor of $n!$ in my sum in the previous.2012-02-08