XLPack Solver – Second order Ordinary Differential Equations (ODEs)

Compute the solution of an initial value problem of a second order ordinary differential equations.

Let’s solve the following example equations.

d2y1/dt2 = -y1/r, d2y2/dt2 = -y2/r (where r = (y1^2 + y2^2)^(3/2))
The initial values are y1 = 1, y2 = 0, dy1/dt = 0 and dy2/dt = sqrt(3) at t0 = 0.

This is an example of a system of second-order simultaneous ordinary differential equations with two dependent variables and two equations.

The solver program will write the value of t into the independent variable (t) cell (B8 in this case) and the values of yi into the dependent variable (yi) cells (B9 to B10 in this case). The differential equation (dyi/dt) cells (B13 and B14 in this case) must contain the formula to compute d2y1/dt2 and d2y2/dt2 from t and yi. To solve this example, the formulas for B13 and B14 are =-B9*(B9^2+B10^2)^(-3/2) and =-B10*(B9^2+B10^2)^(-3/2), respectively.

The initial values and output range (D8:H28 in this case) is the cell range for the value of t (input) in the first column and the computed values of yi and dyi/dt in the second, third, … columns (output). Input the initial values (t0, y1, y2, dy1/dt and dy2/dt at t0 in this case) in the first row and the target t value in the first column of second and later rows, and click “Compute”. Then the values of y1, y2, dy1/dt and dy2/dt at t of each row will be computed and output to the second and later columns of each row. If the row with a blank cell in the first column is encountered, the computation will be terminated even within the initial value and output range.

Dopn43, Dopn64, Dopn86 or Dopn1210 can be selected as the solver program.

The standard value of tolerance is 1.0e-8. The tolerance value will be set to RTol and ATol of each routine.

The cell ranges can be specified as larger than required. In that case, only the necessary range will be used from left upper corner.

Please refer to here for “Save/Restore” button.

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The “?” button of right upper corner will not work correctly. Please use “Help” button.