I have a generating set of differential equations: $$ I_1=u_t\\ I_2=-uu_{tt}\\ I_3=u^{-1}\\ I_4=uu_{xxx}-6u^2u_x $$ where $u=u(x,t)$. and the coordinate is $(x,t,u,u_x,u_t,u_{tt},u_{xxx})$. Infact we assume partitial derivatives of $u(x,t)$ as dependent variables. Therefore, the differential equations become algebraic polynomials. We can regard the set as a Gröbner basis.
I know that the above equations generate the equation $$ u_t+u_{xxx}-6uu_x=0 $$
I want to express eq2 as a function relation of the generating set. which is $$ I_1+I_3I_4=0 $$
I want to know how could I express an equation in terms of generators?
Is there any method or computer package to solve my problem?