I have y ′ = 9 x , h = 1 2 , y ( 0 ) = 1And I want to find y(2) using the improved euler's method. And I know that it goes: y n + 1 = y n + h f ( x n , y n ) + f ( x n + 1 , u n + 1 ) 2 What I don't really "get" is the u. Is it a sort of an replacement for y? In which case it should not mean anything considering the equation I'm given.

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I have  y  ′  =  9  x  ,  h  =      1    2    ,  y  (  0  )  =  1And I want to find y(2) using the improved euler&#039;s method. And I know that it goes:      y          n      +      1        =      y    n    +  h            f      (              x        n            ,              y        n            )      +      f      (              x                  n          +          1                    ,              u                  n          +          1                    )        2  What I don&#039;t really &quot;get&quot; is the   u. Is it a sort of an replacement for   y? In which case it should not mean anything considering the equation I&#039;m given.

spornya1 · Accepted Answer

Lets write it in a form that will clear this up.      y          n      +      1        =      y    n    +            h      2            [    f    (          x      n        ,          y      n        )    +    f    (          x      n        +    h    ,          y      n        +    h    f    (          x      n        ,          y      n        )    )    ]  In the second half of the formula in brackets, we substitute:1.   x  =      x    n    +  h2.   y  =      y    n    +  h  f  (      x    n    ,      y    n    )When we are calculating the iterations.We can also find the exact solution to this DEQ to compare against. The closed form solution is:  y  (  x  )  =            1      2        (  9      x    2    +  2  )

y′=9x,h=1/2,y(0)=1 And I want to find y(2) using the improved euler's method.

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