Engineering Mathematics: November 2014

Thursday, 13 November 2014

Properties of Laplace Transforms

\begin{matrix} 1 . Linearity Property \\ If a, b are any constants and f, g are any functions of t \\ then, \\ L [af (t) + bg (t)] = aL [f (t)] + bL [g (t)] \\ 2 . First Shifting Property \\ If L [f (t)] = f (s) then, \\ L [e^{at} f (t)] = f (s - a) \\ 3 . Change of Scale Property \\ If L [f (t)] = f (s) then, \\ L [f (at)] = \frac{1}{a} f (\frac{s}{a}) \end{matrix}

Laplace Transforms

\begin{matrix} Let f (t) be a function of t defined for all positive values of t . \\ Then the Laplace Transform of f (t) is, \\ L \{f (t)\} = \int_{0}^{\infty} e^{- st} f (t) dt \\ s is a parameter which may be real or complex number . \\ Transforms of Elementary Functions \\ L (1) = \frac{1}{s} proof \\ L (t^{n}) = \frac{n!}{s^{n + 1}} proof \\ L (e^{at}) = \frac{1}{s - a} proof \\ L (\sin at) = \frac{a}{s^{2} + a^{2}} proof \\ L (\cos at) = \frac{s}{s^{2} + a^{2}} proof \\ L (\sinh at) = \frac{a}{s^{2} - a^{2}} proof \\ L (\cosh at) = \frac{s}{s^{2} - a^{2}} proof \\ Properties of Laplace Transforms \\ Linearity \\ First                            Shifting \\ Change                            of                            Scale \\ Solved Problems \\ \begin{matrix} 1 . e^{at} t^{n} & solution \\ 2 . e^{at} \sin bt & solution \\ 3 . e^{at} \cos bt & solution \\ 4 . e^{at} \sinh bt & solution \\ 5 . e^{at} \cosh bt & solution \\ 6 . \sin 2 t \sin 3 t & solution \\ 7 . \cos^{2} 2 t & solution \\ 8 . \sin^{3} 2 t & solution \\ 9 . e^{- 3 t} (2 \cos 5 t - 3 \sin 5 t) & solution \\ 10 . e^{2 t} \cos^{2} t & solution \\ 11 . \sqrt {te}^{3 t} & solution \\ 12 . f (t) = \{\begin{matrix} t / τ, when 0 < t < τ \\ 1, when t > τ \end{matrix} & solution \\ 13 . f (t) = \{\begin{matrix} 1, 0 < t \leq 1 \\ t, 1 < t \leq 2 \\ 0, t > 2 \end{matrix} & solution \\ 14 . {(\sqrt t - \frac{1}{\sqrt t})}^{3} & solution \\ 15 . \frac{\cos \sqrt t}{\sqrt t} & solution \\ 16 . \frac{\sin at}{t} & solution \\ 17 . e^{2 t} + 4 t^{3} - 2 \sin 3 t + 3 \cos 3 t & solution \\ 18 . 1 + 2 \sqrt t + \frac{3}{\sqrt t} & solution \\ 19 . 3 \cosh 5 t - 4 \sinh 5 t & solution \\ 20 . \cos (at + b) & solution \\ 21 . {(\sin t - \cos t)}^{2} & solution \\ 22 . \sin 2 t \cos 3 t & solution \\ 23 . \sin \sqrt t & solution \\ 24 . \sin^{5} t & solution \\ 25 . \cos^{3} 2 t & solution \\ 26 . e^{- at} \sinh bt & solution \\ 27 . e^{2 t} (3 t^{5} - \cos 4 t) & solution \\ 28 . e^{- 3 t} \sin 5 t \sin 3 t & solution \\ 29 . e^{- t} \sin^{2} t & solution \\ 30 . e^{2 t} \sin^{4} t & solution \\ 31 . \cosh at \sin at & solution \end{matrix} \\ \begin{matrix} 32 . \sinh 3 t \cos^{2} t & solution \\ 33 . t^{2} e^{2 t} & solution \\ 34 . {(1 + {te}^{- t})}^{3} & solution \\ 35 . t \sqrt (1 + \sin t) & solution \\ 36 . f (t) = \{\begin{matrix} 4, 0 \leq t \leq 1 \\ 3, t > 1 \end{matrix} & solution \\ 37 . f (t) = \{\begin{matrix} \sin t, 0 < t < π \\ 0, t > π \end{matrix} & solution \\ 38 . f (x) = \{\begin{matrix} \sin (x - π / 3), x > π / 3 \\ 0, x < π / 3 \end{matrix} & solution \\ 39 . f (t) = \{\begin{matrix} \cos (t - 2 π / 3), t > 2 π / 3 \\ 0, t < 2 π / 3 \end{matrix} & solution \\ 40 . f (t) = \{\begin{matrix} t^{2}, 0 < t < 2 \\ t - 1, 2 < t < 3 \\ 7, t > 3 \end{matrix} & solution \end{matrix} \end{matrix}

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