Abstract
Many problems of science and engineering require the description of some measurable quantity (position, temperature, population, concentration, electric current, etc.) as a function of time. Frequently, the scientific laws governing such quantities are best expressed as equations that involve the rate at which that quantity changes over time. Such laws give rise to differential equations.
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- 1.
In this text, we will generally use the prime notation, that is, y′, y′, y′ (and y (n) for derivatives of order greater than 3) to denote derivatives, but the Leibnitz notation \(\frac{\mathrm{d}y} {\mathrm{d}t}\), \(\frac{{\mathrm{d}}^{2}y} {\mathrm{d}{t}^{2}}\), etc. will also be used when convenient.
- 2.
Recall that the standard notations from calculus used to describe an interval I are (a, b), [a, b), (a, b], and [a, b] where a < b are real numbers. There are also the infinite length intervals ( − ∞, a) and (a, ∞) where a is a real number or ± ∞.
- 3.
In practice, this is just implicit differentiation.
- 4.
Technically, we are treating y = y(t) as a function of t and both sides are integrated with respect to t, but the left-hand side becomes an integral with respect to y using the change of variables y = y(t), dy = y′dt.
- 5.
The formula for solving a cubic equation is known as Cardano’s formula after Girolamo Cardano (1501–1576), who was the first to publish it.
- 6.
N(t) could represent the number of atoms or the mass of radioactive material at time t.
- 7.
There are limitations, however. The ratio of 14C to other forms of carbon in the atmosphere is not constant as originally supposed. This variation is due, among other things, to changes in the intensity of the cosmic radiation that creates 14C. To compensate for this variation, dates obtained from radiocarbon laboratories are now corrected using standard calibration tables.
- 8.
A special case was discussed in Exercise 2 of Sect. 1.1.
- 9.
This conflicts with the use of the term standard form, given in Sect. 1.1, where we meant a first order differential equation written in the form y′ = F(t, y). Nevertheless, in the context of first order linear differential equations, we will use the term standard form to mean an equation written as in (1).
- 10.
Unfortunately, the term “homogeneous” used here is the same term used in Sect. 1.4 to describe a linear differential equation in which the forcing function is zero. These meanings are different. Usually, context will determine the appropriate meaning.
- 11.
Named after Jakoub Bernoulli (1654–1705).
- 12.
This equation is known as Clairaut’s theorem (after Alexis Clairaut (1713–1765)) on the equality of mixed partial derivatives.
- 13.
A proof of this theorem can be found in G.F. Simmons’ book Differential Equations with Applications and Historical Notes, 2nd edition McGraw-Hill, 1991.
- 14.
Uniform convergence is defined as follows: for all ε > 0, there exists n 0 such that the maximal distance between the graph of the functions y n (t) and the graph of y(t) (for t ∈ [a′,b′]) is less than ε for all n ≥ n 0 . We will not explore in detail this kind of convergence, but we will note that it implies pointwise convergence. That is, for each t ∈ [α′,β′] y n (t) → y(t).
- 15.
For a proof see, for example, A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Chap. 3, Sect. 11, Dover 1975.
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© 2012 Springer Science+Business Media New York
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Adkins, W.A., Davidson, M.G. (2012). First Order Differential Equations. In: Ordinary Differential Equations. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3618-8_1
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