Answer to Question #221321 in Chemical Engineering for Mohsin Al Mamun

Question #221321

Find the Laplace Transform of the following functions:

i)ft=e-tsinh 3t

                                                       ii)ft=t9e5t


1
Expert's answer
2021-07-30T04:18:01-0400

i

Applytransformrule:ifL{f(t)}=F(s)thenL{eatf(t)}=F(sa)Foretsinh(3t):f(t)=sinh(3t),a=1=L{sinh(3t)}(s+1)=3s29=3(s+1)29\mathrm{Apply\:transform\:rule:\quad if\:}L\left\{f\left(t\right)\right\}=F\left(s\right)\mathrm{\:then\:}\:L\left\{e^{at}f\left(t\right)\right\}=F\left(s-a\right)\\ \mathrm{For\:}e^{-t}\sinh \left(3t\right):\quad f\left(t\right)=\sinh \left(3t\right),\:\quad a=-1\\ =L\left\{\sinh \left(3t\right)\right\}\left(s+1\right)\\ =\frac{3}{s^2-9}\\ =\frac{3}{\left(s+1\right)^2-9}

ii

UseLaplaceTransformtable:L{tkf(t)}=(1)kdkdsk(L{f(t)})Fort9e5t:f(t)=e5t,k=9=(1)9d9ds9(L{e5t})L(e5t)=1s5d9ds9(1s5)=362880(s5)10=(1)9(362880(s5)10)=362880(s5)10\mathrm{Use\:Laplace\:Transform\:table}:\quad \:L\left\{t^kf\left(t\right)\right\}=\left(-1\right)^k\frac{d^k}{ds^k}\left(L\left\{f\left(t\right)\right\}\right)\\ \mathrm{For\:}t^9e^{5t}:\quad f\left(t\right)=e^{5t},\:\quad \:k=9\\ =\left(-1\right)^9\frac{d^9}{ds^9}\left(L\left\{e^{5t}\right\}\right)\\ L({e^{5t}})=\frac{1}{s-5}\\ \frac{d^9}{ds^9}(\frac{1}{s-5})=-\frac{362880}{\left(s-5\right)^{10}}\\ =\left(-1\right)^9\left(-\frac{362880}{\left(s-5\right)^{10}}\right)\\ =\frac{362880}{\left(s-5\right)^{10}}


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