ClearAll[Fx10, Fy10, Fx12, Fy12, x1, y1, x2, y2, t, T1, T2, m1, m2, I1, I2, L1, L2, θ1, θ2, eq1, eq2, eq3, eq4, eq5, eq6,eqns]


eq1 = -Fx10[t] - Fx12[t] == m1*x1''[t];
eq2 = +Fy10[t] - Fy12[t] - m1*g == m1*y1''[t];
eq3 = +T1[t] - T2[t] + (Fx10[t] - Fx12[t])*L1/2*Cos[θ1[t]] - (Fy10[t] + Fy12[t])*L1/2*Sin[θ1[t]] ==I1*θ1''[t];

eq4 = +Fx12[t] == m2*x2''[t];
eq5 = +Fy12[t] - m2*g == m2*y2''[t];
eq6 = +T2[t] - Fx12[t]*L2/2*Cos[θ1[t] + θ2[t]] - Fy12[t]*L2/2*Sin[θ1[t] + θ2[t]] == I2*(θ1''[t] + θ2''[t]);

eqns = {eq1, eq2, eq3, eq4, eq5, eq6};

(*   Code for finding  Fx10,Fy10,Fx12,Fy12   *)
{Fx10[t], Fy10[t], Fx12[t], Fy12[t]} = Solve[eqns, {Fx10[t], Fy10[t], Fx1[t], Fy12[t]}]

(*   Code for finding  Deq1,Deq2             *)
{Deq1, Deq2} = Eliminate[eqns, {Fx10[t], Fy10[t], Fx12[t], Fy12[t]}]