"Which is the most accurate way to solve these equations?"
Generally, the most accurate way to solve any equation is analytically, if it can be solved analytically.
Using symbolic math toolbox -
syms u(t) v(t) w(t) A B E F
eq1 = diff(u,t)+A*B*w(t)*sin(F*t)==0;
eq2 = diff(v,t)==0;
eq3 = diff(w,t)-sin(F*t)*u(t)-A*E*sin(F*t)==0;
con1 = u(0)==0;
con2 = v(0)==0;
con3 = w(0)==0;
sol=dsolve([eq1 eq2 eq3],[con1 con2 con3]);
U(t)=sol.u
V(t)=sol.v
W(t)=sol.w
