# Physics and Linear Differential Equations

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

• – Qmechanic Jan 29 '14 at 14:03
• Because they're easy to solve and produce results with acceptable precision and accuracy. When they don't, we're stuck with PDEs or worse. – Carl Witthoft Jan 29 '14 at 14:27
• @RoopamSinha: We can turn any non-linear differential equation into a set of coupled linear differential equations by a choice of transformations. So the answer is really because we choose to do so. – Kyle Kanos Jan 29 '14 at 15:48
• @joshphysics It does require the transformation about an equilibrium point. Linearization is a very common process of solving non-linear differential equations. – Kyle Kanos Jan 29 '14 at 17:11
• @KyleKanos Ah ok well I'm familiar with linearization in that context, but your original statement is much stronger than that. If linearization is what you're referring to, then I think it's a bit misleading to say "because we choose to do so." There are situations in which the full, nonlinear equation is of interest. – joshphysics Jan 29 '14 at 17:14

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.