I have a 14 equation system of second order differential equations which I am trying to solve using dsolve. The program just keeps running without finishing ?

This is the script: A is 14x14 matrix

syms x1(t) x2(t) x3(t) x4(t) x5(t) x6(t) x7(t) x8(t) x9(t) x10(t) x11(t) x12(t) x13(t) x14(t);
X = [x1;x2;x3;x4;x5;x6;x7;x8;x9;x10;x11;x12;x13;x14];
syms E;
syms alpha;
f = [(k/m) * E * cos(alpha*t);0;0;0;0;0;0;0;0;0;0;0;0;0];
odes = diff(X,2) == A*X + f;
%odes = diff(X,2) == A*X;
%V = odeToVectorField(odes)
%M = matlabFunction(V,'vars',{'t','E','alpha','Y'})
[x1Sol(t),x2Sol(t),x3Sol(t),x4Sol(t),x5Sol(t),x6Sol(t),x7Sol(t),x8Sol(t),x9Sol(t),x10Sol(t),x11Sol(t),x12Sol(t),x13Sol(t),x14Sol(t)] = dsolve(odes);
x1Sol(t) = simplify(x1Sol(t))
x2Sol(t) = simplify(x2Sol(t))
x3Sol(t) = simplify(x3Sol(t))
x4Sol(t) = simplify(x4Sol(t))
x5Sol(t) = simplify(x5Sol(t))
x6Sol(t) = simplify(x6Sol(t))
x7Sol(t) = simplify(x7Sol(t))
x8Sol(t) = simplify(x8Sol(t))
x9Sol(t) = simplify(x9Sol(t))
x10Sol(t) = simplify(x10Sol(t))
x11Sol(t) = simplify(x11Sol(t))
x12Sol(t) = simplify(x12Sol(t))
x13Sol(t) = simplify(x13Sol(t))
x14Sol(t) = simplify(x14Sol(t))

Would really appreciate help on this.

[darova]

The code doesn't work

[Tapas Peshin]

Here you go:

m = 15000;
k = 150000;

n = 14;
A = zeros(n);
A(n,n) = -k/m;
A(n,n-1) = k/m;
for i=1:n-1
    A(i,i) = -2*k/m;
    A(i+1,i) = k/m;
    A(i,i+1) = k/m;
end

disp(A);
syms x1(t) x2(t) x3(t) x4(t) x5(t) x6(t) x7(t) x8(t) x9(t) x10(t) x11(t) x12(t) x13(t) x14(t);
X = [x1;x2;x3;x4;x5;x6;x7;x8;x9;x10;x11;x12;x13;x14];
syms E;
syms alpha;
f = [(k/m) * E * cos(alpha*t);0;0;0;0;0;0;0;0;0;0;0;0;0];
odes = diff(X,2) == A*X + f;
%odes = diff(X,2) == A*X;
%V = odeToVectorField(odes)
%M = matlabFunction(V,'vars',{'t','E','alpha','Y'})
[x1Sol(t),x2Sol(t),x3Sol(t),x4Sol(t),x5Sol(t),x6Sol(t),x7Sol(t),x8Sol(t),x9Sol(t),x10Sol(t),x11Sol(t),x12Sol(t),x13Sol(t),x14Sol(t)] = dsolve(odes);
x1Sol(t) = simplify(x1Sol(t))
x2Sol(t) = simplify(x2Sol(t))
x3Sol(t) = simplify(x3Sol(t))
x4Sol(t) = simplify(x4Sol(t))
x5Sol(t) = simplify(x5Sol(t))
x6Sol(t) = simplify(x6Sol(t))
x7Sol(t) = simplify(x7Sol(t))
x8Sol(t) = simplify(x8Sol(t))
x9Sol(t) = simplify(x9Sol(t))
x10Sol(t) = simplify(x10Sol(t))
x11Sol(t) = simplify(x11Sol(t))
x12Sol(t) = simplify(x12Sol(t))
x13Sol(t) = simplify(x13Sol(t))
x14Sol(t) = simplify(x14Sol(t))

[darova]

What about numerical solution?

[Tapas Peshin]

Couldn't find it. That is why using matlab.

Answer 1

[darova]

alpha and E should be numerical

function main
m = 15000;
k = 150000;
n = 14;
A = zeros(n);
A(n,n) = -k/m;
A(n,n-1) = k/m;
for i=1:n-1
    A(i,i) = -2*k/m;
    A(i+1,i) = k/m;
    A(i,i+1) = k/m;
end
f = zeros(n,1);
x0 = ones(2*n,1);
ts = [0 5];
[t,X] = ode45(myode,ts,x0);
plot(t,X)
    function dx = myode(t,x)
        dx = zeros(2*n,1);
        f(1) = k/m * E * cos(alpha*t);
        dx(1:n) = x(n+1:end);
        dx(n+1:end) = A*x(1:n) + f;
    end
end

[Tapas Peshin]

Question:

1. In my case alpha, E and t are unknowns so I am using symbolic values ?
2. It is a second order differential equation system. Isn't your solution with ode45 for first order ?

Thanks for the help.

[darova]

1. Do you have additional constraints for alpha and E?
2. No. It's for 2d order