In MATLAB, ode45 solver can be used to solve such equations numerically. Since you are solving these equations symbolically, the following correct the error in your code
% finding force
F0=5;
T=1;
t=0:0.01:1;
F1=-F0*(t/T)+F0;
plot(t,F1)
hold on
t2=1:0.01:2
F2=F0*(t2/T)-F0;
plot(t2,F2)
hold on
t3=3:0.01:10;
F3=F0;
F2=[F1,F2,F3];
hold off
% solving problem
syms x1(t) x2(t) F1 F2
m1=1;m2=2;c1=0;c2=0;k1=4;k2=4;k0=0.5;F1=F1;F2=F3;
eq1=F2==m2*(diff(x2,2)-diff(x1,2))+c2*(diff(x2,1)-diff(x1,1))+k2*(x2-x1);
eq2=F1==m1*diff(x1,2)+c1*diff(x1,1)+k1*x1;
Dx1=diff(x1);Dx2=diff(x2);
[X1(t),X2(t)]=dsolve([eq1 eq2],[x1(0)==0,Dx1(0)==0,x2(0)==0,Dx2(0)==0]);
t1=0:0.01:10;
Force=F2;
x1(t)=subs(X2(t),{t,F2,F1},{t1,Force,0});
x2(t)=subs(X2(t),{t,F2,F1},{t1,Force,0});
plot(t1,x1(t), t1,x2(t));legend('X1(t)','X2(t)')
But the two variables x1, and x2 overlap. I am not sure if this is correct solution. If you can give the mathematical form of your equations, I can see if your equations are correctly implemented.
