Differential equations can be used to examine patrials of higher rank with varying coefficients in various regions of the Cartesian coordinate plane. Meanwhile, the researchers and scientists have N. Rajabov, A.S. Star and F.A. Nasim Adeeb Haneen, and others. As a result, while the coefficients of
partial differential equations differ from those of partial differential equations, this research examined the partial differential equation based on its rank (fourth rank). Conditions are established for the production of their coefficients within the context of that equation. In multiple different scenarios involving these coefficients, a single solution for that partial differential equation. These circumstances were summed up in five theories.