quadratic equation extracting square root,Uniqueness of forcing terms in linear partial differential equations,algebra substitution method calculator,how do you divide and times add subtract integers worksheet,multiple choice exponents algebra powerpoint rules,how to solve multivariable algebra equation with fractions,prealgebra solving equations by multiplying and dividing worksheet,adding and subtracting positive and negative integers free worksheets,rules for adding variables in a square root,how do you do the square root on the TI-83 plus graphic calculator?,pre algebra distributive property prentice hall,pre algebra definitions open algebraic expression,linear equation in two variables subject to linear constraint inequalities,solving second order linear difference equation,ti-83 plus solving systems of linear equations in three variables,TI- 83 ,84 graphing calculator step by step of how to use to find slope of lines graphing calculator,finding the least common denominator for two rational expressions,adding subtracting multiplying dividing integers,Quadratic Equation Calculator factor,how to solve second order linear nonhomogeneous differential equations,combining like terms in algebraic expressions worksheets,solving quadratic equations completing the square,pre algebra 6th grade chart line problems,error 13 dimension on a ti 86 graphing calculator solving linear equations,adding subtracting dividing multiplying integers games,multiplying,dividing,adding,subtracting integers,convert decimal to square root fraction,math poems with the words, prime numbers, common multiples,common factors,least common multiple using the ladder method,quadratic equation graph hyperbola,7th grade level variables and equations explanations Equations Substitution variables,What are sqaure root method of Quadratic Equations?,solving nonlinear differential equations matlab,Adding, subtracting, multiplying and dividing Integer worksheets,worksheet on adding,subtracting,multiplying,dividing integers,adding, subtracting, multiplying, dividing integers,test for adding and subtracting negitive and positive numbers,solving second order homogeneous differential equations,Simplifying a sum of radical expressions calculator,online factoring of complex quadratic equations with variables only,free advanced algebra add subtract rational expressions,multiply and divide rational expressions calculator,integers adding, subtracting, multiplying, dividing worksheet,worksheets on add subtract multiply divide fractions,slow steps balancing chemical equations with least common multiples,converting base 2 fraction to decimal fraction,Like Terms Simplifying Algebraic Expressions Activities Lessons,value for variable expression radicals roots,simplify square root of difference of two squares,simplifying exponents and square root calculator,slow steps for balancing chemical equations with least common multiples,factoring polynomials with a cubed term, tutorial,Like Terms Simplifying Algebraic Expressions Junior High,Pre-Algebra chapter 2 evaluate expressions, worksheet triangle expressions answers,converting base 8 to decimal calculator,solving basic algebra equations homework word problem,adding and subtracting fractions with like denominators worksheet,solving NONLINEAR simultaneous equATIONS USING MATLAB,solving multiple variable polynomial equation,Java method Convert Decimal Numbers to time,greatest common factor for three number with variables,great common divisor formula javascript,dividing fractions and mix numbers cheat problem solver,practice worksheets on adding, subtracting,multiplying,and dividing decimals. for 6th grade,convert mixed fraction to decimal,solving absolute value and radical equations using restrictions,quadratic equations square root property calculator
Thank you for visiting our site! You landed on this page because you entered a search term similar to this: Matlab solve second order differential equations, here's the result:

ode23, ode45


Solve ordinary differential equations.


[t,x] = ode23('xprime',t0,tf,x0)[t,x] = ode23('xprime',t0,tf,x0,tol,trace)[t,x] = ode45('xprime',t0,tf,x0)[t,x] = ode45('xprime',t0,tf,x0,tol,trace)


ode23 and ode45 are functions for the numerical solution of ordinary differential equations. They can solve simple differential equations or simulate complex dynamical systems.

A system of nonlinear differential equations can always be expressed as a set of first order differential equations:

where t is (usually) time, x is the state vector, and f is a function that returns the state derivatives as a function of t and x.

ode23 integrates a system of ordinary differential equations using second and third order Runge-Kutta formulas. ode45 uses fourth and fifth order formulas.

The input arguments to ode23 and ode45 are

-------------------------------------------------------------------xprime  A string variable with the name of the M-file that defines           the differential equations to be integrated. The function            needs to compute the state derivative vector, given the cur          rent time and state vector. It must take two input argu              ments, scalar t (time) and column vector x (state), and              return output argument xdot, a column vector of state de             rivatives:                                                   -------------------------------------------------------------------

-------------------------------------------------------------------t0     The starting time for the integration (initial value of t).    tf     The ending time for the integration (final value of t).        x0     A column vector of initial conditions.                         tol    Optional - the desired accuracy of the solution. The default          value is 1.e-3 for ode23, 1.e-6 for ode45.                   trace  Optional - flag to print intermediate results. The default            value is 0 (don't trace).                                      -------------------------------------------------------------------


Consider the second order differential equation known as the Van der Pol equation:

You can rewrite this as a system of coupled first order differential equations:

The first step towards simulating this system is to create a function M-file containing these differential equations. Call it vdpol.m:

function xdot = vdpol(t,x)xdot = [x(1).*(1-x(2).^2)-x(2); x(1)]
Note that ode23 requires this function to accept two inputs, t and x, although the function does not use the t input in this case.

To simulate the differential equation defined in vdpol over the interval 0 t ode23:

t0 = 0; tf = 20;x0 = [0 0.25]';  % Initial conditions[t,x] = ode23('vdpol',t0,tf,x0);plot(t,x)
The general equations for a dynamical system are sometimes written as

where u is a time dependent vector of external inputs and y is a vector of final measured outputs of the system. Simulation of equations of this form are accomplished in the current framework by recognizing that u(t) can be formed within the xdot function, possibly using a global variable to hold a table of input values, and that y = g(x,t) is simply a final function application to the simulation results.


ode23 and ode45 are M-files that implement algorithms from [1]. On many systems, MEX-file versions are provided for speed.

ode23 and ode45 are automatic step-size Runge-Kutta-Fehlberg integration methods. ode23 uses a simple second and third order pair of formulas for medium accuracy and ode45 uses a fourth and fifth order pair for higher accuracy.

Automatic step-size Runge-Kutta algorithms take larger steps where the solution is more slowly changing. Since ode45 uses higher order formulas, it usually takes fewer integration steps and gives a solution more rapidly. Interpolation (using interp1) can be used to give a smoother time history plot.


If ode23 or ode45 cannot perform the integration over the full time range requested, it displays the message

Singularity likely.

See Also

odedemo, SIMULINK


[1] G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, 1977.

(c) , Inc.