help ode45
ODE45 Solve non-stiff differential equations, medium order method.
[TOUT,YOUT] = ODE45(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates
the system of differential equations y' = f(t,y) from time T0 to TFINAL
with initial conditions Y0. ODEFUN is a function handle. For a scalar T
and a vector Y, ODEFUN(T,Y) must return a column vector corresponding
to f(t,y). Each row in the solution array YOUT corresponds to a time
returned in the column vector TOUT. To obtain solutions at specific
times T0,T1,...,TFINAL (all increasing or all decreasing), use TSPAN =
[T0 T1 ... TFINAL].
[TOUT,YOUT] = ODE45(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default
integration properties replaced by values in OPTIONS, an argument created
with the ODESET function. See ODESET for details. Commonly used options
are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector
of absolute error tolerances 'AbsTol' (all components 1e-6 by default).
If certain components of the solution must be non-negative, use
ODESET to set the 'NonNegative' property to the indices of these
components.
ODE45 can solve problems M(t,y)*y' = f(t,y) with mass matrix M that is
nonsingular. Use ODESET to set the 'Mass' property to a function handle
MASS if MASS(T,Y) returns the value of the mass matrix. If the mass matrix
is constant, the matrix can be used as the value of the 'Mass' option. If
the mass matrix does not depend on the state variable Y and the function
MASS is to be called with one input argument T, set 'MStateDependence' to
'none'. ODE15S and ODE23T can solve problems with singular mass matrices.
[TOUT,YOUT,TE,YE,IE] = ODE45(ODEFUN,TSPAN,Y0,OPTIONS) with the 'Events'
property in OPTIONS set to a function handle EVENTS, solves as above
while also finding where functions of (T,Y), called event functions,
are zero. For each function you specify whether the integration is
to terminate at a zero and whether the direction of the zero crossing
matters. These are the three column vectors returned by EVENTS:
[VALUE,ISTERMINAL,DIRECTION] = EVENTS(T,Y). For the I-th event function:
VALUE(I) is the value of the function, ISTERMINAL(I)=1 if the integration
is to terminate at a zero of this event function and 0 otherwise.
DIRECTION(I)=0 if all zeros are to be computed (the default), +1 if only
zeros where the event function is increasing, and -1 if only zeros where
the event function is decreasing. Output TE is a column vector of times
at which events occur. Rows of YE are the corresponding solutions, and
indices in vector IE specify which event occurred.
SOL = ODE45(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be
used with DEVAL to evaluate the solution or its first derivative at
any point between T0 and TFINAL. The steps chosen by ODE45 are returned
in a row vector SOL.x. For each I, the column SOL.y(:,I) contains
the solution at SOL.x(I). If events were detected, SOL.xe is a row vector
of points at which events occurred. Columns of SOL.ye are the corresponding
solutions, and indices in vector SOL.ie specify which event occurred.
Example
[t,y]=ode45(@vdp1,[0 20],[2 0]);
plot(t,y(:,1));
solves the system y' = vdp1(t,y), using the default relative error
tolerance 1e-3 and the default absolute tolerance of 1e-6 for each
component, and plots the first component of the solution.
Class support for inputs TSPAN, Y0, and the result of ODEFUN(T,Y):
float: double, single
See also ODE23, ODE113, ODE15S, ODE23S, ODE23T, ODE23TB, ODE15I,
ODESET, ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT, DEVAL,
ODEEXAMPLES, RIGIDODE, BALLODE, ORBITODE, FUNCTION_HANDLE.
Documentation for ode45
doc ode45
help dsolve
DSOLVE Symbolic solution of ordinary differential equations.
DSOLVE will not accept equations as strings in a future release.
Use symbolic expressions or sym objects instead.
For example, use syms y(t); dsolve(diff(y)==y) instead of dsolve('Dy=y').
DSOLVE(eqn1,eqn2, ...) accepts symbolic equations representing
ordinary differential equations and initial conditions.
By default, the independent variable is 't'. The independent variable
may be changed from 't' to some other symbolic variable by including
that variable as the last input argument.
The DIFF function constructs derivatives of symbolic functions (see sym/symfun).
Initial conditions involving derivatives must use an intermediate
variable. For example,
syms x(t)
Dx = diff(x);
dsolve(diff(Dx) == -x, Dx(0) == 1)
If the number of initial conditions given is less than the
number of dependent variables, the resulting solutions will obtain
arbitrary constants, C1, C2, etc.
Three different types of output are possible. For one equation and one
output, the resulting solution is returned, with multiple solutions to
a nonlinear equation in a symbolic vector. For several equations and
an equal number of outputs, the results are sorted in lexicographic
order and assigned to the outputs. For several equations and a single
output, a structure containing the solutions is returned.
If no closed-form (explicit) solution is found, then a
warning is given and the empty sym is returned.
DSOLVE(...,'IgnoreAnalyticConstraints',VAL) controls the level of
mathematical rigor to use on the analytical constraints of the solution
(branch cuts, division by zero, etc). The options for VAL are TRUE or
FALSE. Specify FALSE to use the highest level of mathematical rigor
in finding any solutions. The default is TRUE.
DSOLVE(...,'MaxDegree',n) controls the maximum degree of polynomials
for which explicit formulas will be used in SOLVE calls during the
computation. n must be a positive integer smaller than 5.
The default is 2.
DSOLVE(...,'Implicit',true) returns the solution as a vector of
equations, relating the dependent and the independent variable. This
option is not allowed for systems of differential equations.
DSOLVE(...,'ExpansionPoint',a) returns the solution as a series around
the expansion point a.
DSOLVE(...,'Order',n) returns the solution as a series with order n-1.
Examples:
% Example 1
syms x(t) a
dsolve(diff(x) == -a*x) returns
ans = C1/exp(a*t)
% Example 2: changing the independent variable
x = dsolve(diff(x) == -a*x, x(0) == 1, 's') returns
x = 1/exp(a*s)
syms x(s) a
x = dsolve(diff(x) == -a*x, x(0) == 1) returns
x = 1/exp(a*s)
% Example 3: solving systems of ODEs
syms f(t) g(t)
S = dsolve(diff(f) == f + g, diff(g) == -f + g,f(0) == 1,g(0) == 2)
returns a structure S with fields
S.f = (i + 1/2)/exp(t*(i - 1)) - exp(t*(i + 1))*(i - 1/2)
S.g = exp(t*(i + 1))*(i/2 + 1) - (i/2 - 1)/exp(t*(i - 1))
syms f(t) g(t)
v = [f;g];
A = [1 1; -1 1];
S = dsolve(diff(v) == A*v, v(0) == [1;2])
returns a structure S with fields
S.f = exp(t)*cos(t) + 2*exp(t)*sin(t)
S.g = 2*exp(t)*cos(t) - exp(t)*sin(t)
% Example 3: using options
syms y(t)
dsolve(sqrt(diff(y))==y) returns
ans = 0
syms y(t)
dsolve(sqrt(diff(y))==y, 'IgnoreAnalyticConstraints', false) warns
Warning: The solutions are subject to the following conditions:
(C67 + t)*(1/(C67 + t)^2)^(1/2) = -1
and returns
ans = -1/(C67 + t)
% Example 4: Higher order systems
syms y(t) a
Dy = diff(y);
D2y = diff(y,2);
dsolve(D2y == -a^2*y, y(0) == 1, Dy(pi/a) == 0)
syms w(t)
Dw = diff(w);
D2w = diff(w,2);
w = dsolve(diff(D2w) == -w, w(0)==1, Dw(0)==0, D2w(0)==0)
See also SOLVE, SUBS, SYM/DIFF, odeToVectorfield.
Documentation for dsolve
doc dsolve
