In order to obtain invariable physical laws, we have to proceed to differential equations, showing the direction of change at each moment, not the integral change after a finite interval, however short.
If the laws of perspective were sufficiently known, the connection between different aspects would be expressed in differential equations.
In a perfected science, causal laws will take the form of differential equations--or of finite-difference equations, if the theory of quanta should prove correct.
In mathematics I find Chances, Figure of the Earth with variable density, Differential Equations, Partial Differentials, sketch for an instrument for shewing refraction, and Optical instruments with effects of chromatic aberration.
But among the remaining parts of Pure Mathematics we have the theory of Elliptic Functions and of the Jacobian and Abelian Functions, and the theory of Differential Equations, including of course Partial Differential Equations.
It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations.
It has been the final aim of Lie from the beginning to make progress in the theory of differential equations; as subsidiary to this may be regarded both his geometrical developments and the theory of continuous groups.
We have seen that the coordinates of bodies are determined by differential equations of the second order, and that so are the differences of these coordinates.
From this new point of view, the distances will be determined by differential equations of the second order.
Mathematicians would say the movements of all the material molecules of the universe depend on differential equations of the second order.
Suppose we observe n molecules and ascertain that their 3n coordinates satisfy a system of 3n differential equations of the fourth order (and not of the second order as the law of inertia would require).
The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.
Differential equations of the first order with two variables.
Integration of Differential Equations of the First Order.
Solution of various problems in geometry which lead to differential equations of the first order.
Differential equations of the same deduced from their characteristic geometrical properties.
Footnote 8: The same error was afterward committed, in the infancy of the infinitesimal calculus, in relation to the integration of differential equations.
Differential equations arise in the expression of the relations between quantities by the elimination of details, either unknown or regarded as unessential to the formulation of the relations in question.
The above list will hopefully provide you with a few useful examples demonstrating the appropriate usage of "differential equations" in a variety of sentences. We hope that you will now be able to make sentences using this group of words.
