solve second order ode system numerically

hello I'm trying to solve this system of second order ordinary differential equations using ode functions (not dsolve):

k1*x' + k2*y' + k3*x + k4*y = u
k5*x" + k6*y" + k7*y' + k8*y = 0

is there a way to convert this system to first order system of equations like in "odeToVectorField"? also is there a function that acts inversely? (like a "VectorFieldToode") to combine the two equations into one higher order differential equation.

Answer 1

[Star Strider]

I would normally expect this to work for other systems of differential equations:

syms k1 k2 k3 k4 k5 k6 k7 k8 u x(t) y(t) Y 
Dx = diff(x);
D2x = diff(Dx);
Dy = diff(y);
D2y = diff(Dy);
Eqn1 = k1*Dx + k2*Dy + k3*x + k4*y == u;
Eqn2 = k5*D2x + k6*D2y + k7*Dy + k8*y == 0;
[VF,Subs] = odeToVectorField(Eqn1, Eqn2)
odeFcn = matlabFunction(VF, 'Vars',{t, Y, u, [k1, k2, k3, k4, k5, k6, k7, k8]})

However, that code throws this error:

Error using mupadengine/feval (line 187)
Number of indeterminates exceeds the number of ODEs.

I must leave it to you to resolve that problem.

[hosein Javan]

yes, that's strange. I think this function is unable to solve it. I did some mathematical manipulation to change the form of equations to first order and it worked:

syms k1 k2 k3 k4 k5 k6 k7 k8 u Du x(t) y(t) z(t) w(t) Y
Dx = diff(x);
Dy = diff(y);
Dz = diff(z);
Dw = diff(w);
Eqn1 = Dx == z;
Eqn2 = Dy == w;
Eqn3 = k1*Dz + k2*Dw + k3*z + k4*w == Du;
Eqn4 = k5*Dz + k6*Dw + k7*w + k8*y == 0;
[VF,Subs] = odeToVectorField(Eqn1, Eqn2, Eqn3, Eqn4)

thank you! if you could, please inform mathworks to find the bug.

[Star Strider]

As always, my pleasure!

Actually, it is not a bug. You changed your system of differential equations, and the condition that threw the original error was no longer present.

[Maxime Sabet D'Acre]

Hi So this thread was never solved?