Physics and Linear Differential Equations

Question

Why in physics, most of the physical systems are modelled by linear differential equations?

Answer 1

I think your qualification of "most" systems needs some clarification because really almost all of the classical universe is described by second-order, nonlinear partial differential equations. Fluids/liquids/gases and solids are described by the same set of second-order, nonlinear PDE's.

Linear equations, both linear PDE's and linear ODE's, show up often because they are a simplified approach to describe something in a tractable way. Nonlinear equations are very difficult to analyze or solve; linear ones are not nearly as hard. Nonlinear equations are difficult to approximate numerically; linear ones are not nearly as hard. So whenever possible, simplifications to things will be made to reduce them to linear equations provided the simplifications are justified and don't ignore fundamental physics (ie. give terrible answers).

Additionally, linear equations allow superposition of solutions. In fluids, for instance, the flow around a cylinder is very difficult to solve using the full Navier-Stokes equations. But if you reduce it to the linear potential flow equations, the solution is just the sum of a uniform flow solution and a doublet solution (or an irrotational vortex solution if the cylinder is spinning). So you can begin to build complicated solutions to complicated geometries and problems iff the equations are linear.

As to why we use differential equations at all -- because we choose to. Many times we can use integral equations or integro-differential equations, all for the same physical problem. The choice depends on your methods of analysis and what you are trying to capture from the equations, but all are valid options.

Answer 2

Differential equations describe a relationship in which the rate of change of some variables are related to the value of other variables. The reason they are so ubiquitous in physics is that this is how nature works. As an example, consider Fouirer's law for heat flux in a material in one dimension; $$ \frac{d q}{dt}=-k\frac{d T}{dx}. $$ This equation can be expressed in words as the rate of heat flow at a given point in an object is equal to a constant (known as the thermal conductivity) times the gradient of the temperature. This law can be expressed in words such as I have just done, but the mathematical expression (which is much simpler to read once you are used to it) is the differential equation written above.