. Recall that, for the Taylor method of order 2, the function T

(2)(t, y) is defined as

T

(2)(t, y) = f(t, y) + h

2

âˆ‚f

âˆ‚t (t, y) + h

2

âˆ‚f

âˆ‚y (t, y)f(t, y), (1)

where we have used the chain rule to evaluate the derivative term. We want to derive a RungeKutta method by approximating T

(2)(t, y) in the form

a1f(t, y) + a2f(t + Î±1, y + Î²1f(t, y)), (2)

where a1, a2, Î±1, and Î²1 are parameters that we must choose.

(a) Taylorâ€™s theorem in two variables implies that we may write

f(t + Î±1, y + Î²1f(t, y)) = f(t, y) + Î±1

âˆ‚f

âˆ‚t (t, y) + Î²1f(t, y)

âˆ‚f

âˆ‚y (t, y) + R1(t + Î±1, y + Î²1),

where R1(t + Î±1, y + Î²1) is the remainder term. If we want (2) to approximate (1), then we

must choose the parameters a1, a2, Î±1, and Î²1 such that the matching condition

f(t, y)+

h

2

âˆ‚f

âˆ‚t (t, y)+

h

2

f(t, y)

âˆ‚f

âˆ‚y (t, y) = a1f(t, y)+a2

f(t, y)+Î±1

âˆ‚f

âˆ‚t (t, y)+Î²1f(t, y)

âˆ‚f

âˆ‚y (t, y)

holds. By matching the coefficients on like-terms (color-coded for your convenience), derive

the three equations that involve the parameters a1, a2, Î±1, and Î²1 (but do not solve these

equations). Make sure to distribute a2 into the bracketed expression before matching!

(b) You will notice that there are four unknown parameters but only three equations, which

suggests that we have some flexibility in choosing the parameters. The standard approach

is to choose a2 freely and then solve for the remaining three parameters a1, Î±1, and Î²1. Do

so for the choice a2 = 1, and state the name of the Runge-Kutta method from lecture

that corresponds to these parameters.

(c) Now choose a2 =

2

3

and solve for the remaining parameters a1, Î±1, and Î²1. This choice of

parameters gives Ralstonâ€™s method. (Note: There is another method where you choose

a2 =

3

4

that is also called Ralstonâ€™s method, but we will go with the one above.