# Time derivative of parameters within ODE solvers

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Edited: Torsten on 23 Nov 2018
I have an ODE which has a parameter whose 1st and 2nd order time derivatives are also included in the ODE:
function dy = ODE(t,y)
f1 = myfun(y(t));
dy = y + f1 + df1/dt + ddf1/dt^2;
end
Unfortunately, the function of analytical derivatives of myfun is not available. Therefore, df1 and ddf1 can be computed numerically, only. Given that the time step in Matlab ode solvers is not fixed, I wonder if there is a way to numerically compute df1 and ddf1.

Torsten on 23 Nov 2018
Edited: Torsten on 23 Nov 2018
function dy = ODE(t,y)
dt = 1e-8;
fm = myfun(t-dt);
f = myfun(t);
fp = myfun(t+dt);
df = (fp - fm) / (2 * dt);
ddf = (fp - 2 * f + fm) / dt^2;
dy = y + f + df + ddf;
end

Torsten on 23 Nov 2018
Does "myfun" depend explicitly on t or only via y ? I.e. can myfun look like y^2+t^2 or only like y^2 ?
It depends on y only, i.e: y^2.
Torsten on 23 Nov 2018
Let
z = f(y)
the value that "myfun" returns for argument y.
Then
dz/dt = df/dy * dy/dt
d^2z/dt^2 = d^2f/dy^2 * (dy/dt)^2 + df/dy * d^2y/dt^2
Inserting into your differential equation gives
dy/dt = y + f + df/dy * dy/dt + d^2f/dy^2 * (dy/dt)^2 + df/dy * d^2y/dt^2
or
df/dy * d^2y/dt^2 + (df/dy - 1) * dy/dt + d^2f/dy^2 * (dy/dt)^2 + y + f = 0
Now you can approximate df/dy and d^2f/dy^2 as I suggested above and solve the system (z1 = y, z2 = dy/dt)
z1' = z2
z2' = -((df/dy - 1) * z2 + d^2f/dy^2 * (z2)^2 + z1 + f)/(df/dy)
using ODE45, e.g.

R2017b

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