Question: solve ode where unknowns have a dependent variable

Hello,

I am trying to solve the following system of odes where the unknowns are xhat, yhat, uhat and vhat. They depend on the variables (x, y, u, v, a_1, a_2, a_3, a_4, f(x)) where f(x) is an arbitrary function and I want to solve it with respect to a_1. The RHS has f(x) and its derivative with respect to x. The dsolve function fails to solve the ode when I add f(x) at the list of variables. 

restart;
vars := x, y, u, v, a_1, a_2, a_3, a_4, f(x);
sys_ode := diff(xhat(vars), a_1) = a_3*x + a_1, diff(yhat(vars), a_1) = a_2 - a_4*y + f(x), diff(uhat(vars), a_1) = a_3*u + 2*a_4*u, diff(vhat(vars), a_1) = a_4*v + diff(f(x), x)*u;

initvars := x, y, u, v, 0, a_2, a_3, a_4, f(x);
ics := xhat(initvars) = x, yhat(initvars) = y, uhat(initvars) = u, vhat(initvars) = v;
solution_a1 = dsolve([sys_ode, ics], [xhat, yhat, uhat, vhat]);

Can anyone help regarding this issue ?