MATH 308 Lecture 18

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Lecture Notes

Laplace Transform of Derivatives

${\mathcal {L}}\left\{f'\right\}(s)=s\,{\mathcal {L}}\left\{f\right\}(s)-f(0)$

Proof.

${\begin{aligned}{\mathcal {L}}\left\{y'\right\}&=\int _{0}^{\infty }y'(t)\,\mathrm {e} ^{-s\,t}\,\mathrm {d} t\\&=\lim _{N\to \infty }\left.y\,\mathrm {e} ^{-s\,t}\right|_{0}^{N}+s\,\int _{0}^{N}\mathrm {e} ^{-s\,t}y\,\mathrm {d} t\\&={\cancel {\lim _{N\to \infty }y(N)\,\mathrm {e} ^{-s\,N}}}-y(0)+s\,{\mathcal {L}}\left\{y\right\}(s)\end{aligned}}$

Therefore ${\mathcal {L}}\left\{y'\right\}=s\,{\mathcal {L}}\left\{y\right\}(s)-y(0)$

Q.E.D.

In General

${\mathcal {L}}\left\{f^{(n)}\right\}(s)=s^{n}\,{\mathcal {L}}\left\{f\right\}(s)-s^{n-1}\,f(0)-s^{n-2}\,f'(0)-\dots -f^{(n-1)}(0)$

Exercise 2

Find the Laplace Transform of $\mathrm {e} ^{4x}$

${\begin{aligned}f(t)&=\mathrm {e} ^{4t}\\f'(t)&=4\mathrm {e} ^{4t}=4\,f(t)\\{\mathcal {L}}\left\{f'(t)\right\}&=4\,{\mathcal {L}}\left\{f(t)\right\}\\s\,{\mathcal {L}}\left\{f(t)\right\}-4\mathrm {e} ^{4\cdot 0}&=4\,{\mathcal {L}}\left\{f(t)\right\}\\(s-4)\,{\mathcal {L}}\left\{f(t)\right\}&=1\\{\mathcal {L}}\left\{f(t)\right\}&={\frac {1}{s-4}}\end{aligned}}$

Exercise 5

$y''-4y=1\quad y(0)=0\quad y'(0)=1$

${\begin{aligned}{\mathcal {L}}\left\{y''\right\}-4{\mathcal {L}}\left\{y\right\}&={\mathcal {L}}\left\{1\right\}={\frac {1}{s}}\\s^{2}\,{\mathcal {L}}\left\{y\right\}-y'(0)-sy(0)-4{\mathcal {L}}\left\{y\right\}&={\frac {1}{s}}\\(s^{2}-4)\,{\mathcal {L}}\left\{y\right\}-1-0s&={\frac {1}{s}}\\{\mathcal {L}}\left\{y\right\}&={\frac {{\frac {1}{s}}+1}{s^{2}-4}}={\frac {s+1}{s\,(s-2)\,(s+2)}}\end{aligned}}$

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

${\begin{aligned}&={\frac {a}{s}}+{\frac {b}{s-2}}+{\frac {c}{s+2}}\\&=a\,{\mathcal {L}}\left\{1\right\}+b\,{\mathcal {L}}\left\{\mathrm {e} ^{2t}\right\}+c\,{\mathcal {L}}\left\{\mathrm {e} ^{-2t}\right\}\\&={\mathcal {L}}\left\{a+b\,\mathrm {e} ^{2t}+c\,\mathrm {e} ^{-2t}\right\}\\y&=a+b\,\mathrm {e} ^{2t}+c\,\mathrm {e} ^{-2t}\end{aligned}}$

We'll learn how to find coefficients $a$, $b$, and $c$ later.

Quiz Time!