Abstract
The method for solving a first order linear differential equation \(y^{\prime} + p(t)y = f(t)\) (Algorithm 3 of Sect. 5) involves multiplying the equation by an integrating factor μ(t) = e∫p(t) dt chosen so that the left-hand side of the resulting equation becomes a perfect derivative (μ(t)y)′. Then the unknown function y(t) can be retrieved by integration. When p(t) = k is a constant, μ(t) = ekt is an exponential function. Unfortunately, for higher order linear equations, there is not a corresponding type of integrating factor.
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Notes
- 1.
A nice proof of this fact can be found on page 442 of the text Advanced Calculus (second edition) by David Widder, published by Prentice Hall (1961).
- 2.
Technically, f is the function while f(t) is the value of the function f at t. Thus, to be correct, the notation should be ℒ{f}(s). However, there are times when the variable t needs to be emphasized or f is given by a formula such as in ℒ{e2t}(s). Thus, we will freely use both notations: ℒ{f(t)}(s) and ℒ{f(s)}.
- 3.
A rational function is the quotient of two polynomials. A rational function is proper if the degree of the numerator is less than the degree of the denominator.
- 4.
In fact, any function which has a power series with infinite radius of convergence, such as an exponential polynomial, is completely determined by it values on [0, ∞). This is so since \(f(t) ={ \sum }_{n=0}^{\infty }\frac{{f}^{(n)}(0)} {n!} {t}^{n}\) and f (n)(0) are computed from f(t) on [0, ∞).
- 5.
In Chap. 6, we will discuss a so-called “generalized function” that will act as a multiplicative identity for convolution.
- 6.
For a proof, see Theorem 6 and the remark that follows on page 450 of the text Advanced Calculus (second edition) by David Widder, published by Prentice Hall (1961).
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Adkins, W.A., Davidson, M.G. (2012). The Laplace Transform. In: Ordinary Differential Equations. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3618-8_2
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