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

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!

[darova]

• Please will you help me introduce the dependance on t of k(t), thanks!

Can you be more specific? What is it? Where is dependance?

Answer 1

[Star Strider]

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.

[Mark Anslow]

Thank you!

[Star Strider]

As always, my pleasure!

[J. Alex Lee]

As long as you are allowed to make k a function handle, can you simply encode the equation that generates kv and tv?

k = @(t)(exp(t))

For example, would be better than

tv = linspace(0,10,100);
kv = exp(tv);
k = @(t) interp1(tv, kv, t, 'linear', 'extrap');            % Function Interpolating ‘k’ To Specific Time

Or is there something about symbolic toolbox that it won't work this way?

[Star Strider]

@J. Alex Lee —

‘... where k(t) is an array of values ...’

So your idea is not appropriate here.

[J. Alex Lee]

oops guess i need help with reading skills. i saw a similar question before, where "k(t)" in the function sense was being confused with "k(t)" in the index sense. Hope this isn't one of those cases and one really needs the discretized time-dependence.

[Star Strider]

Noted.