Solving a nonhomogeneous ODE system with 'dsolve'

To solve a nonhomogeneous ODE system, I used the next code, but it didn't work and showed the following error messages.

Could you check where is the problem?

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syms ct mt kt cb mb kb mc w a0 xtt(t) xt(t) xbb(t) xb(t) xcc(t) xc(t)

A=[-ct/mt -kt/mt ct/mt kt/mt 0 0;

1 0 0 0 0 0;

ct/mb kt/mb -(ct+cb)/mb -(kt+kb)/mb cb/mb kb/mb;

0 0 1 0 0 0;

0 0 cb/mc kb/mc -cb/mc -kb/mc;

0 0 0 0 1 0];

B=[a0/mt*cos(w*t); 0; 0; 0; 0; 0];

Y=[xtt; xt; xbb; xb; xcc; xc];

odes = diff(Y) == A*Y + B;

[xttSol(t), xtSol(t), xbbSol(t), xbSol(t), xccSol(t), xcSol(t)]=dsolve(odes);

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Error using mupadengine/feval_internal (line 172)

Invalid operand.

Error in dsolve>mupadDsolve (line 328)

T = feval_internal(symengine,'symobj::dsolve',sys,x,options);

Error in dsolve (line 183)

sol = mupadDsolve(args, options);

Error in odesys (line 19)

[xttSol(t), xtSol(t), xbbSol(t), xbSol(t), xccSol(t), xcSol(t)]=dsolve(odes);

Answer 1

[Star Strider]

I have no idea what the problem is with this, however I’ve been running it offline on my computer (using a Laplace transform approach) for about 4 hours and I have no results. I am now convinced that in spite of its apparent simplicity, it does not have a symbolic solution.

Try this instead —

syms ct mt kt cb mb kb mc w a0 xtt(t) xt(t) xbb(t) xb(t) xcc(t) xc(t) t 
A=[-ct/mt -kt/mt ct/mt kt/mt 0 0;
    1 0 0 0 0 0;
    ct/mb kt/mb -(ct+cb)/mb -(kt+kb)/mb cb/mb kb/mb;
    0 0 1 0 0 0;
    0 0 cb/mc kb/mc -cb/mc -kb/mc;
    0 0 0 0 1 0]

A = 

B=[a0/mt*cos(w*t); 0; 0; 0; 0; 0]

B = 

Y=[xtt; xt; xbb; xb; xcc; xc

odes = diff(Y) == A*Y + B;

[VF,Subs] = odeToVectorField(odes)

Eqns_Fcn = matlabFunction(VF)

Y(t) = 

VF = 

Subs = 

Eqns_Fcn = function_handle with value:
    @(Y,a0,cb,ct,kb,kt,mb,mc,mt,t,w)[Y(2);(a0.*cos(t.*w))./mt-(ct.*Y(2))./mt+(ct.*Y(3))./mt-(kt.*Y(1))./mt+(kt.*Y(4))./mt;-((cb+ct).*Y(3))./mb-((kb+kt).*Y(4))./mb+(cb.*Y(5))./mb+(ct.*Y(2))./mb+(kb.*Y(6))./mb+(kt.*Y(1))./mb;Y(3);(cb.*Y(3))./mc-(cb.*Y(5))./mc+(kb.*Y(4))./mc-(kb.*Y(6))./mc;Y(5)]

Use ‘Eqns_Fcn’ with the numeric differential equation integrator (ode45 or ode15s for example, depending on the coefficient values) to integrate it numerically. That call would likely be ‘@(t,Y)Eqns_Fcn(Y,a0,cb,ct,kb,kt,mb,mc,mt,t,w)’. Use ‘string(Subs)’ in the legend argument to appropriately display and label the results.

This is the best I can do (and likely the best MATLAB can do) for this problem.

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