Solving Differential Equations in MATLAB

Description of ode23 from paraphrased from MATLAB HELP (you must type help ode23)

ODE23 integrates a system of ordinary differential equations using 2nd and 3rd order Runge-Kutta formulas.

[T,Y] = ODE23( 'F', [ T0  Tfinal] , Y0 )  integrates the system of ordinary differential equations described by the M-file F.M, over the interval T0 to Tfinal, with initial conditions Y0.  


WHERE:

F - String containing name of user-supplied problem description.

You must create a function file named F.m and put the differential equation function in it

[ t0   Tfinal ] is a row-vector containing:  Initial value of t and final value of t for the interval of interest

Y0 - Initial value column-vector.

T - Returned integration time points (column-vector).

Y - Returned solution, one solution column-vector per tout-value.

The result can be displayed by: plot(tout, yout).

EXAMPLES

1) 1st order linear differential equation

Solve the following DE: dx/dt + 3x + 4 = 0. Rewrite the DE as dx/dt = -3x - 4. To calculate x on the interval 0 £ t £ 3, with initial value of x(0) = 12:

Define a function that describes the differential equation. To do this, use File menu option New to create the function below. Save it as LFO (MATLAB will add .m).

function xdot = LFO(t,x)

xdot= - 3*x - 4; (that’s it... save the function file)

Now submit this function to the ode23 function, which computes ordinary differential equations using 2nd and 3rd order Runge-Kutta methods.

ª [t,x]=ode23('LFO', [0 3], 12);

ª plot(t,x)
 
 
 
 

2) 2nd order linear differential equation

Solve the following differential equation on the interval 0 £ t £ 10:

 d2v/dt2 +2 dv/dt + 3 v = 4

with initial value of v(0) = 12 and v’(0) = 0:

Rewrite the differential equation as a system of first order differential equations: let v(1) = v and v(2) = dv/dt = dv(1)/dt. Then,

v’(2)= -2v(2) - 3v(1) + 4

v’(1) = v(2)

Define a function that describes the differential equation. To do this, use File menu option New to create the function below. Save it as RLC.

function vdot = RLC(t,v)

vdot(2, 1)= -2*v(2) - 3*v(1) + 4;

vdot(1,1) = v(2);          (that’s it... save the function file!)

>> initial = [ 12   0 ]’;

>> [t,v]=ode23( 'RLC' ,  [0 10] ,  initial );

>> plot(t, v) 


You will see TWO functions plotted as functions of t: v and  v'.