MATLAB: MATLAB: solving a 2nd order ODE which has a parameter which evolves in time

2nd orderevolvesMATLABodeode45time

Hi, I'm trying to solve the equation: y'' + k(t)y = 0 where k(t) is an array of values and y(0) = 1/2 and y'(0) = 1 in the range t = 0 to 20.
I understand it is possible to solve this as 2 coupled first order differential equations using ode45 as follows when k is constant:
k = 2
syms y(t)
[System] = odeToVectorField(diff(y, 2) == -(k).*y);
M = matlabFunction(System,'vars', {'t','Y'});
sol = ode45(M,[0 20],[1/2 1]);
fplot(@(x)deval(sol,x,1), [0, 20])
Please will you help me introduce the dependance on t of k(t), thanks!

Best Answer

Try this: 
syms k(t) y(t)                                              % Declare ‘k(t)’ Here
[System,Subs] = odeToVectorField(diff(y, 2) == -(k).*y);    % Note ‘k(t)’ Included In Code
M = matlabFunction(System,'vars', {'t','Y'})
kv = rand(1,20);                                            % ‘k’ Vector
tv = linspace(0, 20, numel(kv));                            % Time Vector Matching ‘k’
k = @(t) interp1(tv, kv, t, 'linear', 'extrap');            % Function Interpolating ‘k’ To Specific Time
M = @(t,Y)[Y(2);-k(t).*Y(1)];                               % Output Of 'matlabFunction’ (Needs To Be Copied, And Pasted In The Code After The 'k' Function Is Defined)
[T,Y] = ode45(M,[0 20],[1/2 1]);
figure
plot(T, Y)
grid
xlabel('T')
ylabel('Amplitude')
legend(string(Subs))
Use your own ‘kv’ vector. The rest of the code should adapt to it, whatever it is.
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