How can we use Euler's method to approximate the solutions for the following IVP below: y ′ = − y + t y 1 / 2 , with 1 ≤ t ≤ 2 , y ( 1 ) = 2 ,and with h = 0.5The main concern is the organization, i.e., set up of it for this particular example.And, if the actual solution to the IVP above is: y ( t ) = ( t − 2 + 2 e ⋅ e − t / 2 ) 2 then, how to compare the actual error and compare the error bound?

Question

How can we use Euler&#039;s method to approximate the solutions for the following IVP below:      y    ′    =  −  y  +  t      y          1              /            2        ,   with   1  ≤  t  ≤  2  ,     y  (  1  )  =  2  ,and with   h  =  0.5The main concern is the organization, i.e., set up of it for this particular example.And, if the actual solution to the IVP above is:  y  (  t  )  =  (  t  −  2  +      2        e    ⋅            e              −      t              /            2            )    2  then, how to compare the actual error and compare the error bound?

arbotsck8sg · Accepted Answer

Fix a small step   h and iterate:      t    0    =  1        y    0    =  y  (  1  )  =  2        t          n      +      1        =      t          n        +  h        y          n      +      1        =      y          n        +  h  (  −      y          n        +      t          n            y          n              1              /            2        )and the value       y          n        ≈  y  (      t    n    )  .Try different small step   h  =  0.1  ,  0.01  ,  …  , and compare for all values of       t    0    ,      t    0    +  h  ,      t    0    +  2  h  ,  … the accuracy between the actual solution   y  (      t    n    ) you have and       y    n   you computed.

Jaiden Bowman · Accepted Answer

Thanks, good

How can we use Euler's method to approximate the solutions for the following IVP below:

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