Lecture 3 MATH 223 (Lt Col Tahmina Sultana)

 

 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:  

$\displaystyle F(t)$	$\displaystyle \mathcal{L}\{F(t)\}= f(s)$
$\displaystyle 1$	$\displaystyle \frac{1}{s}$
$\displaystyle t$	$\displaystyle \frac{1}{s^2}$
$\displaystyle t^2$	$\displaystyle \frac{2}{s^3}$
$\displaystyle t^2$	$\displaystyle \frac{6}{s^4}$
$\displaystyle t^n$	$\displaystyle \frac{n!}{s^{n + 1}}$
$\displaystyle e^{at}$	$\displaystyle \frac{1}{s - a}$
$\displaystyle \sin at$	$\displaystyle \frac{a}{s^2 + a^2}$
$\displaystyle \cos at$	$\displaystyle \frac{s }{s^2 + a^2}$

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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:  $\mathcal{L}\{\cosh at\}$