Second Order Laplace solving doesn't work ('Unable to find explicit solution')

I'm trying to solve an ODE using Laplace method, but I'm stuck on solving the equation

Here's my code:

syms t x(t) s X(s);

% PARAMETERS (tried to do symbolically but it was more diffcult)
m = 1;
k = 0.5;
xi = 1.2;
c = xi*2*sqrt(k*m)

c = 1.6971

f0 = 1;
w = 0.1;

dx = diff(x, t, 1);
ddx = diff(x, t, 2);

% INITIAL CONDITIONS
x0 = 0;
dx0 = 0;

newton = m*ddx+ c*dx +k*x;
f = f0*cos(w*t);

lteqn = laplace(newton, t, s)

lteqn = 

lefteqn = subs(lteqn,{laplace(x(t), t, s), x(0),dx(0)},{X(s), x0, dx0})

lefteqn = 

F_s = laplace(f, t, s);
simplify(solve(lefteqn == F_s, X(s)))

Warning: Unable to find explicit solution. For options, see help.

 
ans =
 
Empty sym: 0-by-1
 

I can't believe MATLAB cannot solve this easy equation. I think I'm missing something.

Thank you guys

Answer 1

[Aquatris]

Not sure the underlying reason behind it but you are defining the X as a function of s and solve function seems to be having trouble solving it. Define X as a standalone and it has no trouble

syms t x(t) s X;

% PARAMETERS (tried to do symbolically but it was more diffcult)
m = 1;
k = 0.5;
xi = 1.2;
c = xi*2*sqrt(k*m)

c = 1.6971

f0 = 1;
w = 0.1;

dx = diff(x, t, 1);
ddx = diff(x, t, 2);

% INITIAL CONDITIONS
x0 = 0;
dx0 = 0;

newton = m*ddx+ c*dx +k*x;
f = f0*cos(w*t);

lteqn = laplace(newton, t, s);
lefteqn = subs(lteqn,{laplace(x(t), t, s), x(0),dx(0)},{X, x0, dx0});
F_s = laplace(f, t, s);
simplify(solve(lefteqn == F_s, X))

ans = 

[Matteo Millone]

Thank you so much, I appreciate it a lot :)

Answer 2

[Sam Chak]

Hi @Matteo Millone

Do you want to analytically solve the ODE like this?

syms s t X

%% original parameters
m   = 1;
k   = 0.5;
xi  = 1.2;
c   = xi*2*sqrt(k*m);
f0  = 1;
w   = 0.1;

%% Test parameters -> should return x(t) = 1/2·(sin(t) - t·e^(-t))
% m   = 1;
% k   = 1;
% xi  = 1;
% c   = xi*2*sqrt(k*m);
% f0  = 1;
% w   = 1;

%% Main
eqn = m*s^2*X + c*s*X + k*X == laplace(f0*cos(w*t), t, s);
X   = solve(eqn, X);
x   = ilaplace(X, s, t)

x = 

[Matteo Millone]

Thanks a lot, anyway i need the solution function of s, i have to draw Bode diagram