d A d t = 0.5 × A × ( 1 − A 100 ) − 10with A ( 0 ) = 70 and we want to use Euler's method to get an approximate value for A ( 10 ), with a step size of 1.So the answer sheet says you basically have to use Ans + 0.5 × Ans × ( 1 − Ans 100 ) − 10 with the first Ans being 70, and then of course repeat 10 times.But I'm wondering, doesn't this actually give you d A ( 10 ) d t ? How is this a correct method?

Question

d        A                    d        t              =  0.5  ×  A  ×  (  1  −            A      100        )  −  10with   A  (  0  )  =  70 and we want to use Euler&#039;s method to get an approximate value for   A  (  10  ), with a step size of 1.So the answer sheet says you basically have to use   Ans  +  0.5  ×  Ans  ×  (  1  −            Ans      100        )  −  10 with the first Ans being 70, and then of course repeat 10 times.But I&#039;m wondering, doesn&#039;t this actually give you                     d        A        (        10        )                    d        t            ? How is this a correct method?

Ricky Lamb · Accepted Answer

The Euler method does not give you             d      A              d      t      . You give it a formula for             d      A              d      t      , such as the one in your question. Then from any given point, like your start of (0,70) it puts a straight line through the point with slope             d      A              d      t       of that point. From your expression,             d      A              d      t                  |              (      0      ,      70      )        =  0.5 so we step one unit in t at a slope of 0.5, giving the A value of the next point as 70+0.5⋅1=70.5. Now we are at (70.5,1), we calculate             d      A              d      t       at this point and take another step along the t axis, and so on until we get to t=10

tonan6e · Accepted Answer

Your formula is wrong. If the differential equation is                     d        A                    d        t              =  f  (  A  ) and your step size is   h, the formula is       A          n      +      1        =      A    n    +  h  f  (      A    n    ).So in this case,       A    n    +  0.5      A    n    (  1  −      A    n        /    100  )  −  10.

dA/dt=0.5A(1−A/100)−10 with A(0)=70 and we want to use Euler's method to get an approximate value for A(10), with a step size of 1.

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