  In quantum field theory, the Dyson-Schwinger equations are an infinite set of
coupled equations relating $n$-point Green's functions in a self-consistent
manner. They have found important applications in non-perturbative studies,
ranging from quantum chromodynamics and hadron physics to strongly correlated
electron systems. However, they are notoriously formidable to solve. One of the
main problems is that a finite truncation of the infinite system is
underdetermined. Recently, Bender et al. [Phys. Rev. Lett. 130, 101602 (2023)]
proposed to make use of the large-$n$ asymptotic behaviors and successfully
obtained accurate results in $D=0$ spacetime. At higher $D$, it seems more
difficult to deduce the large-$n$ behaviors. In this paper, we propose another
avenue in light of the null bootstrap. The underdetermined system is solved by
imposing the null state condition. This approach can be extended to $D>0$ more
readily. As concrete examples, we show that the cases of $D=0$ and $D=1$ indeed
converge to the exact results for several Hermitian and non-Hermitian theories
of the $g\phi^n$ type, including the complex solutions.
