How do I solve a second order ode with time dependent terms?

I am trying to solve the turbulent heat transfer equation and I am not able to make the ode function.

The equation shown above is the one I am trying to solve. I am solving for temperature (T*) and the function f is not explicitly known. I have a matrix which contains the values of f, f' and f'' (the function f, its first and second derivative). The ode function which I made was as follows: (ETA_1 is the independent variable and T_1 is the dependent variable)

function T_out_1 = ODE_1_THERM(ETA_1,i,j,T_1,beta,y_out_1_final,ETAspan_1,Pr,Pr_t)
T_out_1 = zeros(2,1); 
T_out_1(:,1) = T_1(:,2);
T_out_1(:,2) = -(y_out_1_final(:,1,j)+beta(1,j)*ETAspan_1(:,i,j).*y_out_1_final(:,3,j)/Pr_t + beta(1,j)*ETAspan_1(:,i,j).^2.*y_out_1_final(:,4,j)/(2*Pr_t))...
               *T_1(2)/(1/Pr+beta(1,j)*ETAspan_1(:,i,j).^2.*y_out_1_final(:,3,j)/(2*Pr_t));
end

The vector ETAspan contains values of ETA itself and y_out_1_final is the matrix where I have stored the values of the function f and its derivative at different values of ETA corresponding to ETAspan.

Answer 1

[Ameer Hamza]

Your equation seems correct. To introduce a time-varying parameter, see this example: https://www.mathworks.com/help/releases/R2020a/matlab/ref/ode45.html#bu3uhuk. It shows how to use interp1() to use datapoints as time-varying parameters.

[Suraj Krishnamurti]

I figured out my mistake. I really had to use the interpol function to make the number of elements of my parameters compatible with the independent and dependent variables in the equation.

Thanks a lot! :)