MATLAB Mathematics - SERC - Index of

Initial Value Problems for ODEs and DAEs
Working with Higher Order ODEs
The ODE solvers accept only first-order differential equations. However, ODEs
often involve a number of dependent variables, as well as derivatives of order
higher than one. To use the ODE solvers, you must rewrite such equations as
an equivalent system of first-order differential equations of the form
y′ = fty ( , )
You can write any ordinary differential equation
y ( n)
= f( t, y, y′…y , ,
( n – 1)
)
as a system of first-order equations by making the substitutions
y 1 = y,
y 2 = y′ , …, y n = y ( n – 1)
The result is an equivalent system of n
first-order ODEs.
y′ 1 = y 2
y 2 ′ = y 3
y n ′
=…
f( t, y 1 , y 2 ,..., yn)
“Example: Solving an IVP ODE (van der Pol Equation, Nonstiff)” on page 5-9
rewrites the second-order van der Pol equation
2
y′′ 1 – µ ( 1 – y 1 ) y′ 1 + y 1 = 0
as a system of first-order ODEs.
Solvers for Explicit and Linearly Implicit ODEs
This section describes the ODE solver functions for explicit or linearly implicit
ODEs, as described in “Types of Problems Handled by the ODE Solvers” on
page 5-4. The solver functions implement numerical integration methods for
solving initial value problems for ODEs. Beginning at the initial time with
initial conditions, they step through the time interval, computing a solution at
each time step. If the solution for a time step satisfies the solver’s error
tolerance criteria, it is a successful step. Otherwise, it is a failed attempt; the
solver shrinks the step size and tries again.
5-5