Solving coupled ODE symbolically

I need to solve this coupled ODE through laplace transform.

My algorithm

1. I have written the 2nd order ODE in matrix form.
2. Took laplace transform
3. Substituted the initial conditions. (Help Needed: Could not substitute diff(x1(0)) and diff(x2(0)) as zero.
4. Solve the linear equation (Help Needed: Don't know what to do)

I have attached my code.

syms t s x1(t) x2(t) m1 m2 c1 c2 k1 k2 k_c c_c Delta_m f1 f2 omega f_0

M=[m1 0;0 m2]

M = 

C=[c1+c_c -c_c;-c_c c2+c_c]

C = 

K=[k1+k_c -k_c;-k_c k2+k_c]

K = 

F=[f1;f2]

F = 

X=[x1;x2]

X(t) = 

equ=M*diff(X,t,2)+C*diff(X,t)+K*X==F

equ(t) = 

equ=subs(equ,[f1,f2],[f_0*cos(omega*t),0])

equ(t) = 

L1=laplace(equ)

L1 = 

L2=subs(L1,[x1(0),x2(0),diff(x1(0),t,1),diff(x2(0),t,1)],[0 0 0 0]) %Substituting Zero Initial Condition

L2 = 

% In above step the time derivative has to be zero, but it is not.
Sol=solve(L2,[laplace(x1) laplace(x2)])

Error using sym.getEqnsVars>checkVariables (line 98)
Second argument must be a vector of symbolic variables.

Error in sym.getEqnsVars (line 62)
checkVariables(vars);

Error in sym/solve>getEqns (line 429)
[eqns, vars] = sym.getEqnsVars(argv{:});

Error in sym/solve (line 226)
[eqns,vars,options] = getEqns(varargin{:});

Answer 1

[Walter Roberson]

syms t s x1(t) x2(t) m1 m2 c1 c2 k1 k2 k_c c_c Delta_m f1 f2 omega f_0

M=[m1 0;0 m2]

M = 

C=[c1+c_c -c_c;-c_c c2+c_c]

C = 

K=[k1+k_c -k_c;-k_c k2+k_c]

K = 

F=[f1;f2]

F = 

X=[x1;x2]

X(t) = 

equ=M*diff(X,t,2)+C*diff(X,t)+K*X==F

equ(t) = 

equ=subs(equ,[f1,f2],[f_0*cos(omega*t),0])

equ(t) = 

L1=laplace(equ)

L1 = 

dx1 = diff(x1)

dx1(t) = 

dx2 = diff(x2)

dx2(t) = 

lap1 = laplace(x1)

lap1 = 

lap2 = laplace(x2)

lap2 = 

syms LAP1 LAP2
L2 = subs(L1, {x1(0), x2(0), dx1(0), dx2(0), lap1, lap2}, {0, 0, 0, 0, LAP1, LAP2}) %Substituting Zero Initial Condition

L2 = 

Sol = solve(L2, [LAP1, LAP2])

Sol = struct with fields:
    LAP1: (f_0*s*(k2 + k_c + c2*s + c_c*s + m2*s^2))/((omega^2 + s^2)*(k1*k2 + k1*k_c + k2*k_c + c1*m2*s^3 + c2*m1*s^3 + c_c*m1*s^3 + c_c*m2*s^3 + k1*m2*s^2 + k2*m1*s^2 + k_c*m1*s^2 + k_c*m2*s^2 + m1*m2*s^4 + c1*k2*s + c2*k1*s + c1*k_c*s + c_c*k1*s + …
    LAP2: (f_0*s*(k_c + c_c*s))/((omega^2 + s^2)*(k1*k2 + k1*k_c + k2*k_c + c1*m2*s^3 + c2*m1*s^3 + c_c*m1*s^3 + c_c*m2*s^3 + k1*m2*s^2 + k2*m1*s^2 + k_c*m1*s^2 + k_c*m2*s^2 + m1*m2*s^4 + c1*k2*s + c2*k1*s + c1*k_c*s + c_c*k1*s + c2*k_c*s + c_c*k2*s +…

LAP1s = simplify(Sol.LAP1)

LAP1s = 

LAP2s = simplify(Sol.LAP2)

LAP2s = 

%iLap1 = ilaplace(LAP1s)
%iLap2 = ilaplace(LAP2s)

The ilaplace() take longer than this demonstration facility allows

[saravanakumar D]

Thank you