I have the following system: y ′ = y + e x y ( 0 ) = 0The problem asks for applying Euler's method and then finding an expression for the global error. Finally, supposing that lim h − &gt; 0 1 − ( 1 + h e h ) x h x h ( e h − ( 1 + h ) ) = 1Check that the global error tends to 0 as h tends to 0 as well.Let's first apply Euler's method. η 0 = y 0 = 0 x i = x 0 + i h = i h η i + 1 = η i + h f ( x i , η i )Since x i = i h, we have η i + 1 = η i + h ( η i + e i h . Factoring by η i : η i + 1 = η i ( 1 + h ) + h e i h And now I don't how should I proceed. Should I solve the iteration? I've done problems where I can express η i + 1 in terms of η 0 by iterations, but here I can't do so because the h e i h prevents that (or at least I haven't managed to do it).I know that global error is defined as e ( x , h ) = η ( x , h ) − y ( x ) but I don't know how exactly should I use that formula. My guess it that I have to solve the differential equation and use the solution y ( x ), but I don't know how to solve that system or what exactly is η ( x , h ).Any help regarding to solving this problem will be highly appreciated.Thanks!

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I have the following system:      y    ′    =  y  +      e    x    y  (  0  )  =  0The problem asks for applying Euler&#039;s method and then finding an expression for the global error. Finally, supposing that      lim          h      −      &amp;gt;      0                  1      −      (                        1          +          h                          e          h                            )                  x          h                                    x        h            (              e        h            −      (      1      +      h      )      )        =  1Check that the global error tends to 0 as   h tends to 0 as well.Let&#039;s first apply Euler&#039;s method.      η    0    =      y    0    =  0      x    i    =      x    0    +  i  h  =  i  h      η          i      +      1        =      η    i    +  h  f  (      x    i    ,      η    i    )Since       x    i    =  i  h, we have       η          i      +      1        =      η    i    +  h  (      η    i    +      e          i      h      . Factoring by       η    i  :      η          i      +      1        =      η    i    (  1  +  h  )  +  h      e          i      h      And now I don&#039;t how should I proceed. Should I solve the iteration? I&#039;ve done problems where I can express       η          i      +      1       in terms of       η    0   by iterations, but here I can&#039;t do so because the   h      e          i      h       prevents that (or at least I haven&#039;t managed to do it).I know that global error is defined as   e  (  x  ,  h  )  =  η  (  x  ,  h  )  −  y  (  x  ) but I don&#039;t know how exactly should I use that formula. My guess it that I have to solve the differential equation and use the solution   y  (  x  ), but I don&#039;t know how to solve that system or what exactly is   η  (  x  ,  h  ).Any help regarding to solving this problem will be highly appreciated.Thanks!

Christina Matthews · Accepted Answer

The integrating factor for the exact solution is       e          −      x      , so that  (      e          −      x        y  (  x  )      )    ′    =  1    ⟹    y  (  x  )  =  y  (  0  )      e    x    +  x      e    x    .This resonance behavior will make the analysis of the numerical solution a little more complicated.The Euler approximations have the general form       y    i    =  A  (  1  +  h      )    i    +  B      e          i      h      . Inserting into the recursion one finds  B      e    h    =  (  1  +  h  )  B  +  h    ⟹    B  =      h                  e        h            −      1      −      h      and from the initial condition      y    0    =  A  +  B    ⟹        y    i    =  (      y    0    −  B  )  (  1  +  h      )    i    +  B      e          i      h        =      y    0    (  1  +  h      )    i    +            h      (      (      1      +      h              )        i            −              e                  i          h                    )                      e        h            −      1      −      h      Now you can insert the expansions for  (  1  +  h      )    i    =  exp  ⁡  (  i  ln  ⁡  (  1  +  h  )  )  =  exp  ⁡  (  i  h  −            1      2        i      h    2    +            1      3        i      h    3    +  .  .  .  )to find the lower error order terms.of the difference to the exact solution.

I have the following system: y′=y+e^x y(0)=0 The problem asks for applying Euler's method and then finding an expression for the global error.

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