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We use differential equations to model the world around us. For example, the logistic differential equation $$\frac{dx}{dt} = rP\left(1-\frac PK\right)$$

is used to model population. However, it doesn't take into account things like climate, natural disasters, competition among other species, etc. Equations modelling forces (Newton's second law) don't really take into account every force acting on an object (i.e. electrical charge force between surrounding particles).

My question is: How accurate are differential equations really, and to what accuracy can we predict future circumstances and events from them?

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    $\begingroup$ This might be a bit opinion-based of a question, possibly too broad as there are a wide range of DEs found in physics that could be answered in the short format here. $\endgroup$ – Kyle Kanos Apr 28 '15 at 20:29
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    $\begingroup$ Slightly tangential, but Newtons equations DO take into account all forces. That is if you write out all the forces and sum them up, they will equal $ma$. It is up to you to actually write out all the forces. Newtons equations are a physical law; the approximating involved is calculating what those forces are. $\endgroup$ – David Etler Apr 28 '15 at 22:34
  • $\begingroup$ Out of curiosity, why would you offer a biological example system on a physics site? $\endgroup$ – dmckee --- ex-moderator kitten Apr 29 '15 at 3:48
  • $\begingroup$ @dmckee Just wanted to use an example which I thought would be good. :) $\endgroup$ – Tdonut Apr 29 '15 at 3:49
  • $\begingroup$ But it is a bad example of fundamental physical thought, which really tries to pare things down to the simplest essence and is mostly very reliable. That's @DavidEtler's point. Complex systems have to be treated in approximation because they are complicated, not because we don't know the core physics. And in these days of massive, inexpensive clusters even that is getting less and less true. $\endgroup$ – dmckee --- ex-moderator kitten Apr 29 '15 at 3:56
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How accurate are differential equations really, and to what accuracy can we predict future circumstances and events from them?

Why does it matter that they are "differential equations"? Differential equations are just one type of model. The question is how accurate are theoretical models. The answer is, the ones you learn about in high school / college are those that have been found to be "good enough" for at least some practical applications in the past. They are unlikely to represent the state of the art, but they are a useful starting point.

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