Abstract
A differential equation is an equation relating a function with its derivatives. In these equations, the functions often represent physical quantities, the derivatives represent their rates of change and the equation defines their relationship. Differential equations have been and still are a major and important branch of pure and applied mathematics since their invention in the mid-seventeenth century. Differential equations began with the German mathematician Leibniz and the Swiss brother mathematicians Jacob and Johann Bernoulli and some others from 1680 on, not long after Newton’s fluxional equations in the 1670s. Applications were made to geometry, mechanics, and optimization. Most part of the eighteenth century was devoted to the consolidation of the Leibnizian tradition, extending it to several independent variables which led to partial differential equations. New scholars known as experts of mathematics, physics, astronomy, and philosophy, namely Euler, Daniel Bernoulli (the son of Jacob Bernoulli), Lagrange, and Laplace appeared on the scene to solve challenging problems related to theory and applications. Several applications were made to mechanics, particularly to astronomy and continuum mechanics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Definition 8.5.
- 2.
Assume that there exists a second solution z with \(z(x) \ne y(x)\) for some \(x > x_0\). Let \(x_M\ge x_0\) be the largest number such that \(z(\xi ) = y(\xi )\) for all \(\xi \in [x_0,x_M]\). Then the functions y and z yield two different solutions of \(y' = f(x, y)\) through the point \((x_M, y(x_M))\) in some interval centered at \(x_M\), contradicting the theorem. Therefore, such a second solution cannot exist. In the same manner, one shows that the solution is unique for values \(x < x_0\).
- 3.
See Definition 10.3.
- 4.
See Definition 6.1.
- 5.
This is a purely formal computation; while the expression “dy / dx” has a clear mathematical meaning—it stands for the derivative of the function y with respect to x—it is not helpful in the present context to try to attach a meaning to the expressions “dx” and “dy”.
- 6.
See also Chap. 10 on Fourier methods.
- 7.
See Definition 4.4.
- 8.
The minus sign is a convention which is widely adopted in the mathematical theory of the Laplace equation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Brokate, M., Manchanda, P., Siddiqi, A.H. (2019). Differential Equations. In: Calculus for Scientists and Engineers. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-8464-6_11
Download citation
DOI: https://doi.org/10.1007/978-981-13-8464-6_11
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-8463-9
Online ISBN: 978-981-13-8464-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)