Given that x' = tx, x(0) = 1, find x(1), and use Euler’s method with n steps to find an approximation to x(1). (You will end up with a product of n terms whose limit (as n → ∞) is sqrt(e))I calculated x(1) using the standard way of solving the ODE by integration and managed to find sqrt(e) but I'm having trouble finding this using Euler. What I did is:Xn+1 = Xn + X'h where h= 1 N substituting the X' by tx I haveXn+1 = Xn + tx( 1 N ) = Xn(1+ t N )So Xn = (1+ ( 1 + t N ) N and simplifying this I have Xn = ( ( 1 + 1 N ) N ) t and taking the limit as n → ∞ I only get e t . Where did I make a mistake? I don't know where does the square root comes from.

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Given that x&#039; = tx, x(0) = 1, find x(1), and use Euler’s method with n steps to find an approximation to x(1). (You will end up with a product of n terms whose limit (as   n  →  ∞) is sqrt(e))I calculated x(1) using the standard way of solving the ODE by integration and managed to find sqrt(e) but I&#039;m having trouble finding this using Euler. What I did is:Xn+1 = Xn + X&#039;h where h=      1    N  substituting the X&#039; by tx I haveXn+1 = Xn + tx(      1    N  ) = Xn(1+      t    N  )So Xn = (1+  (  1  +      t    N        )    N   and simplifying this I have Xn =   (  (  1  +      1    N        )    N        )    t   and taking the limit as   n  →  ∞ I only get       e    t  . Where did I make a mistake? I don&#039;t know where does the square root comes from.

enracant60 · Accepted Answer

Your approach is more or less correct, but there are some indexing mistakes which may be leading to confusion. Consider for instance that   N is a fixed step size, whereas the update rule applies for any iteration of the algorithm. So the rule is more properly written asXi+1=Xi+tNXi.Now, what is   t at iteration   i?       i    N  , of course. So      X          i      +      1        =      (    1    +          i              N        2              )        X    i  and      X    N    =      ∏          i      =      0              N      −      1            (    1    +          i              N        2              )    .

Given that x' = tx, x(0) = 1, find x(1), and use Euler’s method with n steps to find an approximation to x(1). (You will end up with a product of n terms whose limit (as n -> infty ) is sqrt(e))

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