Significance of differential equations

A differential equation relates a variable to its rates of change. In continuous system simulation it is the main tool for describing how the system state evolves.

• A continuous system has variables that change smoothly with time (position, temperature, voltage, concentration). The rate of change at any moment depends on the current state, which is exactly what a differential equation states.
• It gives the model of the system. For example M x'' + D x' + K x = F(t) models a vehicle suspension and dx/dt = -kx models decay.
• Most practical equations are nonlinear or large and cannot be solved in closed form. They are solved by simulation: numerical integration (Euler, Runge-Kutta) on a digital computer or integrators on an analog computer.
• Once the equation is available we can study transient response, stability and the effect of changing parameters, without experimenting on the real system.
• The same equation form appears in many fields (mechanical, electrical, thermal, biological), so one simulation method serves them all, which is the basis of the analog method.