Concept
Differential equation
Overview
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common in mathematical models and scientific laws; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers, and many numerical methods have been developed to determine solutions with a given degree of accuracy. The theory of dynamical systems analyzes the qualitative aspects of solutions, such as their average behavior over a long time interval.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Differential equation is discussed.
In fact, even if we could integrate every equation of the first order relating to a single function, that would not be sufficient for obtaining the final integral of an equation of any order whatever, inasmuch as every differential equation is not reducible to that of an immediately inferior order. Thus, for example, if we have given any relation between _x_, _y_, _dx/dy_, and _d_²_y_/_dx_², to determine a function…
Auguste Comte, The philosophy of mathematics
states what it is; names the idea; opens on it
The knowledge of the elementary fact enables us to put the problem in an equation. Nothing remains but to deduce from this by combination the complex fact that can be observed and verified. This is what is called _integration_, and is the business of the mathematician. It may be asked why, in physical sciences, generalization so readily takes the mathematical form. The reason is now easy to see. It is not only…
Henri Poincaré, The Foundations of Science
states what it is; names the idea
For this system, which is called that of _occasional causes_ (because it teaches that God acts on the body at the instance of the soul, and _vice versa_), besides introducing perpetual miracles to establish communication [157] between these two substances, does not obviate the derangement of the natural laws obtaining in each of these same substances, which, in the general opinion, their mutual influence would cause.
Gottfried Wilhelm Leibniz, Theodicy Essays on the Goodness of God, the Freedom of Man and the Origin of Evil
states what it is; author linked in the graph
Referring back to the definition (67) for log _ṡ_, we find that the combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish identically, when _f_ and F* are alternating matrices. Accordingly it follows out of {A}, that (68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) = 0, while the relation (69) lor (lor F*) = 0, signifies that of the four equations in {B}, only three represent independent…
Albert Einstein, The Principle of Relativity
names the idea
An important distinction, however, exists between the dynamical and the mathematical regress. The latter is engaged solely with the combination of parts into a whole, or with the division of a whole into its parts; and therefore are the conditions of its series parts of the series, and to be consequently regarded as homogeneous, and for this reason, as consisting, without exception, of phenomena. If the former…
Immanuel Kant, The Critique of Pure Reason
Philosophers who discussed Differential equation
Related ideas
- Gottfried Wilhelm Leibniz
- Springer Science+Business Media
- Isaac Newton
- Function (mathematics)
- Mathematics
- Classical mechanics
- Mechanics
- Heat
- Diffusion
- Physics
- Applied mathematics
- Quantum mechanics
- Quantum field theory
- Statistical mechanics
- Complex system
- Jean Le Rond d'Alembert
- Chemistry
- Symmetry
- Newton's laws of motion
- Chaos theory
- Biology
- Economics
Opposing perspectives
Place in the tradition
Part of Conventionalism.