Single 1st Order ODE

Solving differential equations in MATLAB

ode45

1 3/2/2009 Prepared by Dr.Waleed

2

ode45(‘odefun’,tspan,y0)

tspan:A vector specifying the interval of integration, [t0,tf]

y0:A vector of initial conditions,[y’,y,..]

odefun:A function that evaluates the right side of the differential equations.

3/2/2009 Prepared by Dr.Waleed

1-Re-arrange equation to dy/dt=……….. 2-Specify time interval 3-Specify initial condition

Example1: a single 1st-order ode

Consider solving the following initial value problem:

Step1 :Create the following .m file called ex1.m:

function xprime=ex1(t,x) xprime=sin(t*x);

File name Function

name

10t0for 1, x(0),0)sin()(' txtx


Step2:In the Matlab command window, type: clc

clear

[t,x] = ode45(‘ex1’,[0 ,10], [1]); plot(t,x)

Initial condition Time interval


Time horizontal axis

x vertical axis

Matrix to save IN-OUT

Using ode45 to solve system of 1nd ODE

3/2/2009 Prepared by Dr.Waleed 5

Solving systems of 1st-order odes Consider solving the following system of system of ODE equations:

10t0 and 7.6)0(y ,1.5)0(:

)cos()sin( 0)cos()sin(

)sin()cos(0)sin()cos(

21

1

'

21

'

2

2

'

12

'

1

ywhere

tyytyy

tyytyy


function yprime=ex2(t,z)

Eq1= cos(z(2))+sin(t); Eq2= sin(z(1))-cos(t); yprime=[Eq1; Eq2]; or

yprime=[cos(z(2))+sin(t);sin(z(1))-cos(t)]; Now z and yprime are both column vectors


clc;clear tspan=[0, 10]; y0=[5.1, 6.7]; [t, z]=ode45(‘ex2’, tspan, y0); plot(t, z)

Create new file and type the following:

Of course, you can save the above commands to any name .m file and type the name of the file in the command window to run it.


Using ode45 to solve system of 2nd ODE


Solving 2nd-order ode Consider solving the following 2nd order ODE:

5t0 and 1)0(y ,0)0(:

092

'

'''

ywhere

ytyy

1)0( and 0)0(z

:becomecondition initial and

, ......eq2 29z

....eq1 z

:asn can writte problem

,yz andy z

21

21

'

2

2

'

1

'

21

z

tzz

z

the

Let


function zprime=ex3(t,z) Eq1=z(2) ; Eq2=-9*z(1)-2*t*z(2); zprime=[Eq1;Eq2];


State Variables

Put it in Matrix Form?

State Variables

Put it in Matrix Form?

''' 29 tyyy

Create new file, type:

clc;clear z0=[0,1]; tspan=[0,5]; [t,z]=ode45(‘ex3’, tspan,z0); plot(t,z);


System of 2nd ODE


22322

2121111 )(

xBxKTxM

xxKxKxM

𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑓𝑜𝑢𝑟: 𝑥1′ 0 = ⋯ , 𝑥1 0 = ⋯ 𝑥2′ 0 = ⋯ , x2 0 = ⋯


24

23

12

11

xz

xz

xz

xz


42

12

2332

24

43

31

2121

12

21

)(1

)(1

zM

Bz

M

KzKK

Mz

zz

zM

KzKK

Mz

zz

)4()1()3()(1

)4(

)3()1()(1

)2(

22

232

24

3

1

221

12

1

zM

Bz

M

KzKK

MEq

zEq

zM

KzKK

MEq

zEq

In Matlab form

Single 1st Order ODE

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