To find the characteristic system curve for a two-branch system with different fluids and pumps, a detailed system diagram and specific fluid properties such as density, viscosity, and temperature are essential. The discussion emphasizes the importance of understanding whether the fluids will chemically react or simply mix, with a preference for considering dilution to simplify the analysis. Basic engineering principles, including mass, energy, and momentum conservation equations, must be applied at each node in the system. Additionally, pump curves and friction factor equations are necessary for accurate calculations. The approach to this problem involves both steady-state and transient analysis, highlighting its complexity.