I have differential equations and I need to write them into a vector function

How do you write the right-hand-sides of the differential equations into a vector function?

[Cris LaPierre]

Duplicate of https://www.mathworks.com/matlabcentral/answers/1590454-i-have-initial-value-problem-that-i-have-to-write-the-right-side-into-a-vector-function?s_tid=srchtitle

Answer 1

[William Rose]

@ssmith,

The documentation for ode45() is really good, I think. Can you make an effort at writing some code, and post what you have so far? You already have both first order differential equations, which means you don;t have to do any more math.

A key concept for this kind of problem is that you define a column vector whose elements are the variables for which you have a first order differential equation. In your problem, you could define the column vector x of length 2, where x(1)=the variable you call y, and x(2)=the variable you call z.

You write a function that receives the current value of x and of time t. It returns the column vector of first derivatives of x. You can save the function as its own .m file, or you can include it at the end of your code, after the last line of code. The ode45() documentation decribes all this. YOur function will look like this:

function dxdt = odefun(t,x)
%Inputs
%  x (2x1): x(1)=y, x(2)=z
%  t (scalar)
%Output
%  dxdt (2x1)
dxdt=zeros(2,1); %make sure dxdt is a column and not a row 
dxdt(1)=5 + x(2)/5 - 4*x(1)/(20+3*t);
dxdt(2)=4*x(1)/(20+3*t) - 2*x(2)/5;
end

You'll also need a main program which calls ode45. Again, see the docmentaiton for good examples.

[ssmith]

@William Rose do i just name my equations anything I want and then use the named equations and initial conditions in the ode45()

[William Rose]

@ssmith, Did you edit your original question.? I think you had specified a set of equaitons but now I don;t see it. Can you restore it please? Thank you.

[ssmith]

@William Rose I have fixed it, my mistake sorry

[William Rose]

@ssmith, thanks. I also edited my earlier post.

I had the order of the arguments to odefun() incorrect. I had written odefun(x,t). It should be odefun(t,x), and now it is. I had written x=xeros(2,1) and it should have been dxdt=zeros(2,1). Now it is.

As for what you cal the variables: I used

function dxdt = odefun(t,x)

and therefore I must pass the function name, odefun, to the function ode45() in my main script. I could have called it

function dxdt = myfunc(t,x)

and then I would ave to pass the name myfunc to ode45 in my main script.

I could have specified it as

function dydt = odefun(t,y)

(instead of dxdt) and then the lines of code later in the function declaration would have to assign values to the elements of the output, dydt (instead of dxdt). Specifically,

function dydt = odefun(t,y)
dydt=zeros(2,1); %make sure dydt is a column and not a row 
dydt(1)=5 + y(2)/5 - 4*y(1)/(20+3*t);
dydt(2)=4*y(1)/(20+3*t) - 2*y(2)/5;
end

The name of the initial state vector in the main program can be any legal variable name. It just has to have the number of rows expected by odefun(), which is 2 in this problem. It is conventional and useful, but not required, that the initial state vector name end with zero, and the 0 is removed to name the solution array returned by ode45(). So you might use r (solution array) and r0 (initial state vector) for example.

[ssmith]

@William Rose So I have to create the function dydt in order to write the right hand side as a vector function? I cant do this

w = @(t,y) 5 + (z/5) - ((4*y)/(20+3*t))
w1 = @(t,z) ((4*y)/(20+3*t)) - (2*z)/5
[t,y] = ode45(w, [0, 12], 0)
[t,z] = ode45(w1, [0, 12], 20)

[William Rose]

@ssmith, you define a vector x which encompasses all the first order differential equations in your system. In your case you have 2 first order diff eqs: dy/dt and dz/dt. So let's assume that x(1) is what you call y, and x(2) is what you call z. Then the differential equations are

and

The odefun() which I posted earlier encodes those equations.

In your main, deifne the initial conditions

x0=[0;20];

and the time span of integration

tspan=[0 12];

Then call ode45. I am using a syntax for odefun that assures that teh variables t and x get passed. I honestly thought you could use a simpler synatx, but tonight I am not having success with the simpler syntax, and the following works:

[t,x]=ode45(@(t,x) odefun(t,x),tspan,x0);

ode45() returns a vector t with the times of the solution, from 0 to 12, and it returns an array x with 2 columns: x1 and x2, evaluated at the times listed in vector t. After that you can plot t versus x(:,1) and t versus x(:,2).