@Carl Love Thanks! This is everything I need!

@Carl Love This way I get:

sol := {xk = (-c*xdot - m*xdot2 + F)/k, xk1 = (-dt*k*xdot - c*xdot - m*xdot2 + F)/k, xk2 = (dt^2*k*xdot2 - 2*dt*k*xdot - c*xdot - m*xdot2 + F)/k}

What I would like is only:

xk = (F*dt^2 + c*dt*xk1 + 2*m*xk1 - m*xk2)/(dt^2*k + c*dt + m)

Which is this sol := solve({eq1, eq2, eq3}, {xdot, xdot2, xk}) , but only the variable xk, xdot and xdot2 removed. 

@dharr Great, it works!

@Preben Alsholm Great!

@Preben Alsholm Thanks! This helped me get the right piecewise function.

k := piecewise(t < t1, kmax, And(t1 <= t, t < 2*t1 + sqrt(2)*t1), -kmax, And(2*t1 + sqrt(2)*t1 <= t, t < 3*t1 + 2*sqrt(2)*t1), kmax, And(3*t1 + 2*sqrt(2)*t1 <= t, t < 4*t1 + 2*sqrt(2)*t1), -kmax, 4*t1 + 2*sqrt(2)*t1 <= t, 0)