MATH 223
Lecture 03
Laplace Transformation
Example: Find
\(\mathcal{L}\{\cos at\}\)
Solution: We know that,
$ \displaystyle \int e^{ \alpha .t} \cos \beta t ~dt = \frac{ e^{\alpha t } (\alpha ~ \cos \beta t + \beta \sin \beta t)}{\alpha^2 + \beta^2} $
\begin{align*}
\therefore \mathcal{L}\{\cos at\} = & ~~\\
= & \int^\infty_0 e^{- st} ~ \cos at ~dt \\
= & \left[ \frac{e^{- st}( - s \cos at + a \sin at)}{s^2 + a^2}\right]^\infty_0 \\
= & [\textrm{Do It Yourself }🙄]\\
= & \cdots \cdots~~~\\
= & \frac{s }{s^2 + a^2}
\end{align*}
♦ Some Important Results:
| $ \displaystyle F(t) $ | $ \displaystyle \mathcal{L}\{F(t)\}= f(s) $ |
|---|---|
| $ \displaystyle 1 $ | $ \displaystyle \frac{1}{s}$ |
| $ \displaystyle t $ | $ \displaystyle \frac{1}{s^2}$ |
| $ \displaystyle t^2 $ | $ \displaystyle \frac{2}{s^3}$ |
| $ \displaystyle t^2$ | $ \displaystyle \frac{6}{s^4}$ |
| $ \displaystyle t^n$ | $ \displaystyle \frac{n!}{s^{n + 1}}$ |
| $ \displaystyle e^{at}$ | $ \displaystyle \frac{1}{s - a}$ |
| $ \displaystyle \sin at $ | $ \displaystyle \frac{a}{s^2 + a^2}$ |
| $ \displaystyle \cos at$ | $ \displaystyle \frac{s }{s^2 + a^2}$ |
Hyperbolic Function
$$ \sinh at = \frac{ e^{at} - e^{- at}}{2} $$
$$ \cosh at = \frac{e^{at} + e^{at}}{2} $$
Example: Find
\(\mathcal{L}\{\sinh at\}\)
Solution: By the Definition of Laplace Transformation,
\begin{align*}
\mathcal{L}\{\sinh at\} & = \int^\infty_0 e^{- st} \sinh at ~dt \\
& = \int^\infty_0 e^{- st} \left(\frac{e^{at} - e^{- at}}{2}\right)~dt \\
& = \frac{1}{2} \int^\infty_0 e^{- st} . e^{at} ~dt - \frac{1}{2} \int^\infty_0 e^{- st} . e^{- at} ~dt \\
& = \frac{1}{2} \mathcal{L}\{e^{at}\} - \frac{1}{2} \mathcal{L}\{e^{- at}\} \\
& = \frac{1}{s - a }- \frac{1}{2} \frac{1}{s + a} \\
& = \frac{1}{2} \frac{2a}{s^2 - a^2} \\
& = \frac{a}{s^2 - a^2}
\end{align*}
Home Work:
\(\mathcal{L}\{\cosh at\}\)
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