 Solution

ID: past-exam-of-the-numerical-linear-algebra-and-numerical-solution-of-differential-equations-course-of-the-university-of-oxford/2020/2/a/solution

α_{k} = 1

, has the form

\sum_{j = 0}^{k} α_{j} Y_{n + j} = h \sum_{j = 0}^{k} β_{j} f (x_{n + j}, Y_{n + j}), x_{n} = x_{0} + nh .

Here

Y_{n}

approximates the solution

y (x_{n})

Y_{0}, \dots, Y_{k - 1}

are starting approximations. The coefficients are independent of

n

and

h

. If

β_{k} = 0

, the update is explicit; otherwise it is an implicit time-stepping method, whose new value must satisfy the displayed equation.

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