Consider the initial value problem d y d x = y , y ( 0 ) = 1Approximate y ( 1 ) using Euler's method with a step size of 1 n , where n is an arbitrary natural number. Use this approximation to write Euler's number e as a limit of an expression in n. How large do you have to choose n in order to approximate e up to an error of at most 0.1? Comment on the quality of approximate in this example.What I did is the following: y ( 1 ) ≈ y 1 = y 0 + h f ( x 0 , y 0 ) = 1 + h f ( 0 , 1 ) = 1 + 1 n This is where I stuck, am I on the right direction? What should I do next?

Question

Consider the initial value problem                    d        y                    d        x              =  y  ,  y  (  0  )  =  1Approximate   y  (  1  ) using Euler&#039;s method with a step size of             1      n      , where n is an arbitrary natural number. Use this approximation to write Euler&#039;s number   e as a limit of an expression in   n. How large do you have to choose   n in order to approximate   e up to an error of at most 0.1? Comment on the quality of approximate in this example.What I did is the following:  y  (  1  )  ≈      y    1    =      y    0    +  h  f  (      x    0    ,      y    0    )  =  1  +  h  f  (  0  ,  1  )  =  1  +            1      n      This is where I stuck, am I on the right direction? What should I do next?

Raiden Williamson · Accepted Answer

So, you have found y1 using       x    0   and       y    0  . To find       x    1  , recall that       x    1    =      x    0    +  h  =      1    n   (in general,       x    k    =      x    0    +  k  h  =      x          k      −      1        +  h). Once you find       x    1  , you can find       y    2   by using      y    2    =      y    1    +  h  f  (      x    1    ,      y    1    )  =      (    1    +          1      n        )    +      1    n    f      (          1      n        ,    1    +          1      n        )    =  1  +      2    n    +      1          n      2        =            (      1      +              1        n            )        2    .Can you see that pattern above? What do you think       y    k   will be? (HINT:       y    3    =            (      1      +              1        n            )        3   ) Keep repeating. That is, find       x    n  , then       y          n      +      1        =      y    n    +  h  f  (      x    n    ,      y    n    ). That will help you express   e as a limit.

Consider the initial value problem <mstyle displaystyle="true" scriptlevel="0">

Answered question

Answer & Explanation

New Questions in Integral Calculus