The derivative of the function with respect to y (∂f/∂y) is –4y, and the derivative of the function

with respect to t (∂f/∂t) is

 

− 2t

.

 

+ t2 )2

(1

 

1.Store the independent variable’s initial value, 0, in T.

2.Store the dependent variable’s initial value, 0, in Y.

3.

Store the expression,

 

1

− 2y2

, in F.

1+ t2

 

 

 

 

 

 

4.

Store ∂f/∂y, –4y, in FY.

 

 

5.

Store ∂f/∂t,

 

− 2t

 

, in FT.

 

 

+ t2 )2

 

 

 

(1

 

 

 

 

6. Enter these five items in a list: { T Y F FY FT }.

7.Enter the tolerance. Use estimated decimal place accuracy as a guideline for choosing a tolerance: 0.00001.

8.Enter the final value for the independent variable: 8.

The stack should look like this:

{ T Y F FY FT }

.00001

8

9.Press RRK. The variable T now contains 8, and Y now contains the value .123077277659. The actual answer is .123076923077, so the calculated answer has an error of approximately

.00000035, well within the specified tolerance.

See also: RKF, RKFERR, RKFSTEP, RRKSTEP, RSBERR

RRKSTEP

Type: Command

Description: Next Solution Step and Method (RKF or RRK) Command: Computes the next solution step (hnext) to an initial value problem for a differential equation, and displays the method used to arrive at that result.

The arguments and results are as follows:

•{ list } contains five items in this order:

–The independent variable (t).

–The solution variable (y).

–The right-hand side of the differential equation (or a variable where the expression is stored).

–The partial derivative of y'(t) with respect to the solution variable (or a variable where the expression is stored).

–The partial derivative of y'(t) with respect to the independent variable (or a variable where the expression is stored).

•xtol is the tolerance value.

•h specifies the initial candidate step.

•last specifies the last method used (RKF = 1, RRK = 2). If this is the first time you are using RRKSTEP, enter 0.

•current displays the current method used to arrive at the next step.

•hnext is the next candidate step.

The independent and solution variables must have values stored in them. RRKSTEP steps these variables to the next point upon completion.

Note that the actual step used by RRKSTEP will be less than the input value h if the global error tolerance is not satisfied by that value. If a stringent global error tolerance forces RRKSTEP to reduce its stepsize to the point that the Runge–Kutta–Fehlberg or Rosenbrock methods fails, then

3-212 Full Command and Function Reference