MATH 308 Lecture 18

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Lecture Notes

Laplace Transform of Derivatives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ f' \right\}(s) = s \, \mathcal{L} \left\{ f \right\}(s) - f(0)}

Proof.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathcal{L} \left\{ y' \right\} &= \int_0^\infty y'(t) \, \mathrm{e}^{-s\,t} \, \mathrm{d}t \\ &= \lim_{N\to\infty} \left. y\,\mathrm{e}^{-s\,t} \right|_0^N + s\,\int_0^N \mathrm{e}^{-s\,t} y \,\mathrm{d}t \\ &= \cancel{\lim_{N\to\infty} y(N) \, \mathrm{e}^{-s\,N}} - y(0) + s \, \mathcal{L} \left\{ y \right\} (s) \end{align}}

Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ y' \right\} = s \, \mathcal{L} \left\{ y \right\}(s) - y(0)}

Q.E.D.

In General

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ f^{(n)} \right\} (s) = s^n \, \mathcal{L} \left\{ f \right\} (s) - s^{n-1} \, f(0) - s^{n-2} \, f'(0) - \dots - f^{(n-1)}(0)}

Exercise 2

Find the Laplace Transform of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^{4x}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f(t) &= \mathrm{e}^{4t} \\ f'(t) &= 4\mathrm{e}^{4t} = 4 \, f(t) \\ \mathcal{L} \left\{ f'(t) \right\} &= 4 \, \mathcal{L} \left\{ f(t) \right\} \\ s \, \mathcal{L} \left\{ f(t) \right\} - 4 \mathrm{e}^{4 \cdot 0} &= 4 \, \mathcal{L} \left\{ f(t) \right\} \\ (s-4) \, \mathcal{L} \left\{ f(t) \right\} &= 1 \\ \mathcal{L} \left\{ f(t) \right\} &= \frac{1}{s-4} \end{align}}

Exercise 5

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'' - 4y = 1 \quad y(0) = 0 \quad y'(0) = 1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathcal{L} \left\{ y'' \right\} - 4 \mathcal{L} \left\{ y \right\} &= \mathcal{L} \left\{ 1 \right\} = \frac{1}{s} \\ s^2 \, \mathcal{L} \left\{ y \right\} - y'(0) - sy(0) - 4 \mathcal{L} \left\{ y \right\} &= \frac{1}{s} \\ (s^2 - 4) \, \mathcal{L} \left\{ y \right\} - 1 - 0s &= \frac{1}{s} \\ \mathcal{L} \left\{ y \right\} &= \frac{\frac{1}{s} + 1}{s^2-4} = \frac{s+1}{s\,(s-2)\,(s+2)} \end{align}}

The Laplace transform was good, but the solution is better:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} &= \frac{a}{s} + \frac{b}{s-2} + \frac{c}{s+2} \\ &= a \, \mathcal{L} \left\{ 1 \right\} + b \, \mathcal{L} \left\{ \mathrm{e}^{2t} \right\} + c \, \mathcal{L} \left\{ \mathrm{e}^{-2t} \right\} \\ &= \mathcal{L} \left\{ a + b\,\mathrm{e}^{2t} + c\,\mathrm{e}^{-2t} \right\} \\ y &= a + b \, \mathrm{e}^{2t} + c \, \mathrm{e}^{-2t} \end{align}}

We'll learn how to find coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} later.

Quiz Time!