I am trying to plot this function dxdt=N0*si​n(omega*t)​*x*(1-x/K) to get a 3-D plot but my code does not work,where is the error?

function RunOsciliationsky3D

N0all= 1:1:10;

N=length(N0all);

omegaall= 1:1:10;

M=length(omegaall);

Pmax=zeros(1,N);

Pmean=zeros(1,N);

Pall=[Pmax(i),Pmean(j)];

x=length(Pall);

for i=1:N

    for j =1:N
        [t,x]=ode45(@osciliation,[0 100],0.1,[],N0all(i),10,omegaall(j));
        Pall(i,j)=x;

    end
end
[N0x,omegay]=meshgrid(N0all,omegaall);
h=mesh(N0x,omegay,Pall);

1;

Note: x axis = Noall

      y axis =omegaall

      z axis = Pall, which is a matrix containing the maximum and mean values of x.

Answer 1

[Roger Stafford]

It is not necessary to call on 'ode45' to solve this particular differential equation. By ordinary methods of calculus, integration of both sides of

   dx/(x*(1-x/K)) = N0*sin(omega*t)*dt

will give you

   x/(K-x) = C*exp(-N0/omega*cos(omega*t))

where C is a constant whose value depends on the given initial condition. All you have to do then is put in those initial conditions for x and t to solve for C, and then solve the equation for x to obtain x as an explicit function of t involving the parameters N0 and omega. With this explicit formula you should be able to do whatever plotting you have in mind.

[Avan Al-Saffar]

Dear Roger

Could you gime me more details or simply write the code to me because I am trying but it does not work.

Best regards

Avan

[Roger Stafford]

I will assume you have followed my reasoning via calculus to the equation

 x/(K-x) = C*exp(-N0/omega*cos(omega*t))

where C is a constant parameter whose value depends on the particular initial conditions you have with x. According to your ode45 call, the value of x is to be 0.1 when t = 0. If so, that suffices to determine what C must be:

 0.1/(K-0.1) = C*exp(-N0/omega*cos(omega*0))
             = C*exp(-N0/omega)

Therefore

 C = 0.1/(K-0.1)*exp(N0/omega)

which gives the equation

 x/(K-x) = 0.1/(K-0.1)*exp(N0/omega*(1-cos(omega*t)))

Solving this for x gives:

 x = 0.1*K*exp(N0/omega*(1-cos(omega*t))) / ...
     (K-0.1*(1-exp(N0/omega*(1-cos(omega*t)))))

   = 0.1*K/((K-0.1)*exp(-N0/omega*(1-cos(omega*t)))+0.1)

This last equation is your explicit formula for x as a function of t, derived entirely without the use of ode45. You can do your plotting directly from this formula.

[Avan Al-Saffar]

Thanks a lot for your explanation but actually the problem that I am asking for it is related to the meshgrid, there is something not define well at the beginning but I do not how to deal with it.

Regards