In lecture I derived the formula for the (explicit) Euler method (EE) by expanding the solution y(x) as a Taylor series about x₀ (the left node) and then evaluating the series at x = x₁ (the right node) to obtain

y₁   =   y₀ + h f (x₀, y₀) ,

and a local error of

e₁(h)   =   ½ y''(ξ) h², where ξ is some value of x between x₀ and x₁. (The value of ξ is not arbitrary; it is simply unknown.)

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You will now derive our second method, the implicit Euler method, by performing the following steps.

1. Expand the solution y(x) as a Taylor series about x₁ (the right node) and then evaluate the series at x₀ (the left node) to obtain
   y₀   =   y₁ – h f (x₁, y₁) .

   Then the implicit Euler method is (solve for y₁):

   y₁   =   y₀ + h f (x₁, y₁) .

2. Use the "remainder term" R(x₁) to show that the local error at node 1 (and therefore at each node) is:
   e₁(h)   =   – ½ y''(τ) h² , where τ is some number between x₀ and x₁.

3. Conclude that the global error of the implicit Euler method is
   E(h) = O(h) .

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