Ordinary Differential Equations (basics)
Solve a 2nd-order ODE
The MotionGenesis™ command file below
solves a 2nd-order ODE for x(t).
The command OutputPlot causes ODE() to
Output data and Plot x(t) vs. t (time).
Variable x'' = cos(2*t) + sin(x)
%----------------------------------------------------------
% Set initial values for variables.
Input x = 1 m, x' = 0.2 m/s
%----------------------------------------------------------
% List output quantities (for ODE command and plotting).
OutputPlot t seconds, x meters
%----------------------------------------------------------
% Set numerical integration parameters.
Input tFinal = 20 sec, tStep = 0.1 sec, absError = 1.0E-7
%----------------------------------------------------------
% Solve ODEs and plot data file MGSolve2ndOrderODE.1
ODE() MGSolve2ndOrderODE
Optional: To instead auto-generate a MATLAB® .m file,
change the last line to ODE() MGSolve2ndOrderODE.m
Note: The prime symbol ' denotes differentiation with respect to time t.
Note: The prime symbol ' denotes differentiation with respect to time t.
x'' = cos(2*t) + sin(x)
The output file MGSolve2ndOrderODE.1 can be
plotted with MotionGenesis™ or MATLAB®, etc.
Solve a 1st-order ODE with various initial values.
The MotionGenesis™ command file
below solves an ODE with various initial values.
Variable y' = cos(t) + sin(y) % Arbitrary ODE.
%----------------------------------------------------------------
% List output quantities (for the ODE command and plotting).
Output t sec, y m, y' m/s
%----------------------------------------------------------------
% Set numerical integration parameters.
Input tFinal = 5 sec, tStep = 0.02 sec, absError = 1.0E-7
%----------------------------------------------------------------
% Solve the ODE with initial value y = 3 m.
Input y = 3 m % Set the initial value for y.
ODE() ODEWithInitialValueA % Solve the ODE and output results.
%----------------------------------------------------------------
Input y := 0 m % Modify the initial value of y.
ODE() ODEWithInitialValueB % Solve the ODE and output results.
%----------------------------------------------------------------
% Optional: Plot both ODE results for y(t) on the same graph.
Plot ODEWithInitialValueA.1 [1,2] ODEWithInitialValueB.1 [1,2]
Solve an ODE with on/off if-like statement
The MotionGenesis™ command file below
solves the 1st-order ODE: if( t < 𝛑 ) y' = sin(t) else y' = 0
isTLessThanPi = isPositive(pi-t) % true if t < 𝛑, else 0.
sintOrZero = sin(t) * isTLessThanPi % sin(t) if t < 𝛑, else 0.
Variable y' = sintOrZero
%--------------------------------------------------------------
Input tFinal = 4 sec, y = 0 m % Arbitrary input values.
Output t sec, y meters
%--------------------------------------------------------------
ODE() MGSolveODEWithIfStatement
Plot MGSolveODEWithIfStatement.1 [1, 2] % Plot y vs. time t.
Solve coupled 1st-order ODEs
The MotionGenesis™ command file below
solves ODEs for the angular velocity measures wx, wy, wz of a rotating rigid body.
Constant Ixx = 1 kg*m^2, Iyy = 2 kg*m^2, Izz = 3 kg*m^2
Variable wx', wy', wz'
wx' = (Iyy-Izz) * wy * wz / Ixx
wy' = (Izz-Ixx) * wx * wz / Iyy
wz' = (Ixx-Iyy) * wx * wy / Izz
%----------------------------------------------------------
% Set initial values for variables.
Input wx = 0.2 rad/sec, wy = 7.0 rad/sec, wz = 0.2 rad/sec
%----------------------------------------------------------
% List output quantities (for ODE command and plotting).
OutputPlot t sec, wx rad/sec, wy rad/sec, wz rad/sec
%----------------------------------------------------------
% Set numerical integration parameters.
Input tFinal = 4 sec, tStep = 0.02 sec, absError = 1.0E-7
%----------------------------------------------------------
ODE() MGCoupledODEs
Optional: To instead auto-generate a MATLAB® .m file,
change the last line to ODE() MGCoupledODEs.m
MotionGenesis™ also does vector algebra, matrix algebra and solves sets of linear equations or nonlinear equations.
Advanced:
Solve additional ODEs
and auto-generate MATLAB® C, etc., codes that solve ODEs and take command-line arguments.