how derive a solution of ode45

Hi i have a ode45 solution with this script

syms y(t)

[V] = odeToVectorField(diff(y, 2) + 2*diff(y) - sin(y) == 0)

M = matlabFunction(V,'vars', {'t','Y'})

sol = ode45(M,[0 20],[1 0])

fplot(@(x)deval(sol,x,1),[0, 20])

dsol = diff(sol)

Now i want to derive the solution to obtain the velocity.

How i can do ?

Answer 1

[Torsten]

The red curve is the derivative of the blue curve.

syms y(t)
[V] = odeToVectorField(diff(y, 2) + 2*diff(y) - sin(y) == 0)

V = 

M = matlabFunction(V,'vars', {'t','Y'})

M = function_handle with value:
    @(t,Y)[Y(2);sin(Y(1))-Y(2).*2.0]

sol = ode45(M,[0 20],[1 0])

sol = struct with fields:
     solver: 'ode45'
    extdata: [1×1 struct]
          x: [0 2.3881e-04 0.0014 0.0074 0.0373 0.1865 0.4925 0.8627 1.3143 1.8619 2.5131 3.2315 3.9762 4.7234 5.4770 6.2552 7.0796 7.9773 8.9985 10.3680 11.5660 12.5652 13.5105 14.5711 15.7928 17.2330 18.9829 20]
          y: [2×28 double]
      stats: [1×1 struct]
      idata: [1×1 struct]

hold on
fplot(@(x)deval(sol,x,1),[0, 20])
fplot(@(x)deval(sol,x,2),[0, 20])
hold off

[Luca Tricarico]

Thanks !!

[Luca Tricarico]

So the structure sol has two solutions, the first is the position and the second the velocity?

[Torsten]

If you look at the function handle M, your second-order equation is solved as a system of two first-order equations:

y1' = y2
y2' = sin(y1) -2*y2

Here, y1 is your y und y2 is dy/dt.

So if y1 is position, y2 is velocity, and these are the two components of the sol structure.

This is the disadvantage of OdeToVectorField: the beginner quickly looses control over the solution process.