Ordinary Differential Equations

An ordinary differential equation (ODE) contains one or more derivatives of a dependent variable, y, with respect to a single independent variable, t, usually referred to as time. The notation used here for representing derivatives of y with respect to t is $$y\text{'}$$ for a first derivative, $$y\text{'}\text{'}$$ for a second derivative, and so on. The order of the ODE is equal to the highest-order derivative of y that appears in the equation.

For example, this is a second order ODE:

In an initial value problem, the ODE is solved by starting from an initial state. Using the initial condition, $$y_0$$, as well as a period of time over which the answer is to be obtained, $$\left(t_0,t_f\right)$$, the solution is obtained iteratively. At each step the solver applies a particular algorithm to the results of previous steps. At the first such step, the initial condition provides the necessary information that allows the integration to proceed. The final result is that the ODE solver returns a vector of time steps $$t=\left[t_0,t_1,t_2,\mathrm{...},t_f\right]$$ as well as the corresponding solution at each step $$y=\left[y_0,y_1,y_2,\mathrm{...},y_f\right]$$.

Types of ODEs

The ODE solvers in MATLAB^® solve these types of first-order ODEs:

Systems of ODEs

You can specify any number of coupled ODE equations to solve, and in principle the number of equations is only limited by available computer memory. If the system of equations has n equations,

then the function that encodes the equations returns a vector with n elements, corresponding to the values for $$y{\text{'}}_1,y{\text{'}}_2,\dots ,y{\text{'}}_n$$. For example, consider the system of two equations

A function that encodes these equations is

function dy = myODE(t,y)
  dy(1) = y(2);
  dy(2) = y(1)*y(2)-2;
end

Higher-Order ODEs

The MATLAB ODE solvers only solve first-order equations. You must rewrite higher-order ODEs as an equivalent system of first-order equations using the generic substitutions

The result of these substitutions is a system of n first-order equations

For example, consider the third-order ODE

Using the substitutions

results in the equivalent first-order system

The code for this system of equations is then

function dydt = f(t,y)
  dydt(1) = y(2);
  dydt(2) = y(3);
  dydt(3) = y(1)*y(3)-1;
end

Complex ODEs

Consider the complex ODE equation

where $$y=y_1+i{y}_2$$. To solve it, separate the real and imaginary parts into different solution components, then recombine the results at the end. Conceptually, this looks like

For example, if the ODE is $$y\text{'}=yt+2i$$, then you can represent the equation using the function file:

function f = complexf(t,y)
  f = y.*t + 2*i;
end

Then, the code to separate the real and imaginary parts is

function fv = imaginaryODE(t,yv)
  % Construct y from the real and imaginary components
  y = yv(1) + i*yv(2);            

  % Evaluate the function
  yp = complexf(t,y);             

  % Return real and imaginary in separate components
  fv = [real(yp); imag(yp)]; 
end     

When you run a solver to obtain the solution, the initial condition y0 is also separated into real and imaginary parts to provide an initial condition for each solution component.

y0 = 1+i;
yv0 = [real(y0); imag(y0)];
tspan = [0 2];
[t,yv] = ode45(@imaginaryODE, tspan, yv0);

Once you obtain the solution, combine the real and imaginary components together to obtain the final result.

y = yv(:,1) + i*yv(:,2);

Basic Solver Selection

ode45 performs well with most ODE problems and should generally be your first choice of solver. However, ode23, ode78, ode89 and ode113 can be more efficient than ode45 for problems with looser or tighter accuracy requirements.

Some ODE problems exhibit stiffness, or difficulty in evaluation. Stiffness is a term that defies a precise definition, but in general, stiffness occurs when there is a difference in scaling somewhere in the problem. For example, if an ODE has two solution components that vary on drastically different time scales, then the equation might be stiff. You can identify a problem as stiff if nonstiff solvers (such as ode45) are unable to solve the problem or are extremely slow. If you observe that a nonstiff solver is very slow, try using a stiff solver such as ode15s instead. When using a stiff solver, you can improve reliability and efficiency by supplying the Jacobian matrix or its sparsity pattern.

You can use ode objects to automate solver selection based on properties of the problem. If you are not sure which solver to use, then this table provides general guidelines on when to use each solver.

For details and further recommendations about when to use each solver, see [5].

Summary of ODE Examples and Files

There are several example files available that serve as excellent starting points for most ODE problems. To run the Differential Equations Examples app, which lets you easily explore and run examples, type

odeexamples

To open an individual example file for editing, type

edit exampleFileName.m

To run an example, type

exampleFileName

This table contains a list of the available ODE and DAE example files as well as the solvers and options they use. Links are included for the subset of examples that are also published directly in the documentation.