Use Euler's method with step size 0.2 to estimate y(0.6) where y(x) is the solution of the initial value problem y' = cos(x+y), y(0) = 0So on a high level, we're using Euler's method to approximate a function that we don't know but only know the derivative equation for and the initial. We're going to use Euler's to get approximate values of the unknown function using approximations.So is this right:original x = 0, y = 0 y 1 = 0 + 0.2 ( c o s ( 0 ) ) = 0.2so new x = 0.2 and new y = 0.2 y 2 = 0.2 + 0.2 ( c o s ( 0.2 + 0.2 ) ) ≈ 0.384 y 3 = 0.384 + 0.2 ( c o s ( 0.4 + 0.384 ) ) = 0.384 + 0.141 ≈ 0.525Is that right? The numbers are just going to be hard to work with right?

Question

Use Euler&#039;s method with step size 0.2 to estimate y(0.6) where y(x) is the solution of the initial value problem y&#039; = cos(x+y), y(0) = 0So on a high level, we&#039;re using Euler&#039;s method to approximate a function that we don&#039;t know but only know the derivative equation for and the initial. We&#039;re going to use Euler&#039;s to get approximate values of the unknown function using approximations.So is this right:original   x  =  0,   y  =  0      y    1    =  0  +  0.2  (  c  o  s  (  0  )  )  =  0.2so new   x  =  0.2 and new   y  =  0.2      y    2    =  0.2  +  0.2  (  c  o  s  (  0.2  +  0.2  )  )  ≈  0.384      y    3    =  0.384  +  0.2  (  c  o  s  (  0.4  +  0.384  )  )  =  0.384  +  0.141  ≈  0.525Is that right? The numbers are just going to be hard to work with right?

boomzwamhc · Accepted Answer

Yes you are correct. You are using an explicit Euler method with   h  =  0.2 and each step is calculated via      y    n    =      y          n      −      1        +  h  ⋅      y    ′    (      x          n      −      1        ,      y          n      −      1        )which you did correct. The numbers will be harder to work with but you can either use a computer/calculator to store the &quot;full&quot; number or if specified by the question round the current numbers to a certain factor.The solutions to you problem in each step, using python, are:      y          t      =      0.2        =  0.2        y          t      =      0.4        =  0.38421219880057705        y          t      =      0.6        =  0.5258011762649013As you can see the last number is 0.526 rounded to three decimals which differs by 0.001 form your answer. This might be beacuse you just cut off the last decimals or you rounded in each step and this rounding error propagated forward, explaining this difference.

Use Euler's method with step size 0.2 to estimate y(0.6) where y(x) is the solution of the initial v

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