Abstract
In this chapter, we again consider solving nonhomogeneous linear differential equations such as
but in contexts where the forcing function is different from those we’ve previously encountered. While we have developed the methods of undetermined coefficients and variation of parameters to approach this problem, there are several reasons to consider a different means of solution.
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Notes
- 1.
\(\displaystyle \lim _{r \to \infty } \frac {r}{e^{sr}} = \lim _{r \to \infty } \frac {1}{se^{sr}} = 0\).
- 2.
The integration by parts formula holds since y is continuous. If y has a jump discontinuity, then this part of the argument is more complicated.
- 3.
Technically, the Dirac delta function is not a function, because it has the unusual property that it is zero everywhere but a, and infinite at t = a. Ultimately, the Laplace transform is what enables us to make sense of this function.
- 4.
We know \({\mathcal {L}}^{-1}[1/s] = 1\), and thus the first shifting property implies \({\mathcal {L}}^{-1}[1/(s+1)] = e^{-t} \cdot 1\).
- 5.
More on power series expansions of functions and the meaning of terms such as “analytic” may be found in Sect. 7.2.
- 6.
While the Laplace transform of a finite sum is the sum of the Laplace transforms of the individual terms, it is not obvious that this property holds for infinite sums. The formal justification that this is valid in what follows is beyond the scope of this text; the reader may assume that this step is valid, and proceed as directed.
- 7.
A review of the development of power series of functions can be found in Sect. 7.2.
- 8.
Recall the shortcut \(\left [ \begin {array}{rr} a & b \\ c & d \end {array} \right ]^{-1} = \frac {1}{ad-bc} \left [ \begin {array}{rr} d & -b \\ -c & a \end {array} \right ]\).
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Struthers, A., Potter, M. (2019). Laplace Transforms. In: Differential Equations . Springer, Cham. https://doi.org/10.1007/978-3-030-20506-5_5
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DOI: https://doi.org/10.1007/978-3-030-20506-5_5
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