How can I define this differential equation in matlab syntax to solve it using ode45 ?

I want to solve this equation for tau,

gammadot is not a differential term

the values for lamda1, lambda2, eta0, gamma are known

edit: I just want to know how to write this equation in a way that matlab understands

[Torsten]

Was there something unclear about the responses you already received for this question ?

[Namit Dayal]

Hi Torsten , I apologize, I didnt frame my question correctly, For this equation i just want to understand how to define this equation as there are two differential terms in here and the syntax I've seen is for one term or 2nd order equations with the same differential.

Answer 1

[Sam Chak]

Hi @Namit Dayal

Depending on the values of , this is one way to solve the differential equations using the 'ode45()' solver. Here, I assume that grows linearly with time, as illustrated in the third subplot.

tspan  = [0 1000];
y0     = [1 0 0];
[t, y] = ode45(@odefcn, tspan, y0);

subplot(311)
plot(t, y(:,1)), grid on, ylabel \tau
subplot(312)
plot(t, y(:,2)), grid on, ylabel \gamma
subplot(313)
plot(t, y(:,3)), grid on, ylabel \gamma\prime, xlabel t

%% Differential equations
function dydt = odefcn(t, y)
    dydt    = zeros(3, 1);
    eta0    = 8080;
    lambda1 = 1.109;
    lambda2 = lambda1;
    dydt(1) = - (y(1) + eta0*y(3) + eta0*lambda2*1)/lambda1;
    dydt(2) = t;    % gammadot      = t (grows linearly from 0 to 1000 in 1000 s)
    dydt(3) = 1;    % d gammadot/dt = 1 (differentiate t with respect to t)
end

Answer 2

[Sam Chak]

Hmm... @Namit Dayal, but you declared that "gammadot is not a differential term". Thus, only is regarded as the differential term. Also, the values for the rest of the parameters , , , and are known. If these values are constants such that , then , and the differential equation can be reduced to .

The analytical solution for this first-order ODE is given by

where is the initial value.

I hope you have disclosed all information regarding this math problem. If my assumption about is incorrect, then the solution becomes illegitimate.

[Namit Dayal]

I have a doubt, suppose gamma is a set of values from 0.001 to 1000, then will d(gamma)/dt be equal to zero as well ?

[Sam Chak]

Then, my solution above is 'illegitimate' and needs modification. However, if rich time series data of is available, we can estimate the rate of change of , denoted as .

[Torsten]

I have a doubt, suppose gamma is a set of values from 0.001 to 1000, then will d(gamma)/dt be equal to zero as well ?

In your preceding question that can be found here

https://uk.mathworks.com/matlabcentral/answers/2048307-how-do-i-solve-an-ode-and-plot-a-graph-of-two-terms-within-the-ode?s_tid=prof_contriblnk

I asked

Do you want to compute one function for tau and treat gamma(dot) is a function of t

or

do you want to compute as many functions tau as there are parameters for gamma(dot) and treat gamma(dot) as constant for each of these computations ?

and you answered

I want to treat gamma(dot) as a vector of constants and solve for tau for all values of gamma(dot) separately

Assuming the same is true here, @Sam Chak gave you the solution of the differential equation with gammadot = constant and thus d(gammadot)/dt = 0 (although I doubt this is meant in the differential equation).

[Sam Chak]

Perhaps the OP didn't know how to express the problem to us. 'gammadot' () is indeed a differential term, but it appears as a time-dependent term in data form. If that's the case, then the interpolation method using interp1() can be applied.