Ordinary Differential Equations Solve Blocks

Odesolve([vector], x, b, [step]) Returns a function(s) of x that is the solution to the single ODE provided in a solve block, subject to either initial value or boundary value constraints. The ODE must be linear in its highest derivative term, and the number of initial and boundary conditions must be equal to the order(s) of the ODE(s).

By default, Odesolve uses a fixed-step Runge-Kutta method of solving. To use an adaptive method, right-click on the Odesolve expression and choose Adaptive from the menu. To use a stiff/Radau method, or to solve systems with algebraic constraints, right-click on Odesolve and choose Stiff from the menu.

Odesolve must be used at the end of a Solve Block.

Arguments:

• vector (used only for systems of ODEs) is a vector of function names (with no variable names included) as they appear within the solve block. For example, if you are solving for the functions f(x) and g(x), vector would be

  

  Array subscripts may not be used when naming functions, but literal subscripts are fine.
• x is the name of the variable of integration.
• b is the real terminal point of the integration interval. b must be greater than the initial value, which is inferred from the initial conditions in the Solve Block.
• step (optional) is the integer number of steps used when interpolating the solution from the approximated points. By default, 1000 steps are used.

Notes:

• The universal notes on constructing Solve Blocks apply. Within the body of the block:

  • Unknown functions: Functions must be defined explicitly in terms of their variables, for example, use y(x), not just y.
  • ODE Equations: Equations must be defined using Boolean equals. Equations can be written either using the derivative operators, such as d/dx and d²/dx², or using prime notation, y(x) and y'(x). (The keystroke for the prime character is [Ctrl] [F7].)
  • Initial and boundary conditions: Initial and boundary conditions should be given in the form of y(0) = q or y'(b) = q. In each case, the endpoints used in boundary conditions must match the endpoints specified in the Odesolve command. Mathcad will check for the correct type and number of conditions and give an error if there is a mistake.
  • Constraints: Mathcad accepts algebraic constraints, such as y(b) + z(b) = w(b). Algebraic constraints add an extra unknown function, w, to the system, which must be specified as one of the output functions in Odesolve. Inequality constraints are not allowed.
• Assign the output of the Odesolve function to a function name or vector of function names, without arguments. These functions are understood to have the integration variable as an argument. You can parameterize the solve block, but you then have to assign the output with particular parameter values to another function name, then call that with the integration variable.
• ODE Solve Blocks work by calling the appropriate method as in the command-line solvers. The solution is approximated at the number of points determined by the solver, then interpolated using lspline at uniform intervals, as specified by step. Both the adaptive and stiff methods use non-uniform step sizes, adding more steps in regions of greater variation of the solution; the default value of step has been set large enough that the interpolation should give a reasonable representation of the solution, but you may increase this value to capture fine details in your solution. Doing so requires Odesolve to return more points between which to interpolate and may increase solution time. Reducing the number of steps may reduce calculation time, but also runs the risk of missing details in the solution.
• To solve an ODE that is not linear in the highest derivative term, or to solve ODEs at the command line, as in a program loop, use rkfixed or one of the other command-line ODE solvers. You can also assign the output of a parameterized solve block in a program loop using local functions.
• Depending on the scale of the problem, and the relative step size used, you may need to reduce the value of TOL to get appropriate solutions. Try this when you get an error relating to "too many integrator steps."
• Beware of aliasing when solving problems with periodic behavior over many cycles. Be sure you have sampled with enough interpolation points to get the frequencies in your results you expect.

QuickSheet - Solving First-Order ODEs

QuickSheet - Solving a First-Order System of ODEs

QuickSheet - Simple Pendulum Motion