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anonymous
 one year ago
what is the laplace transform of t^2u(t1)?
anonymous
 one year ago
what is the laplace transform of t^2u(t1)?

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DARTHVADER2900
 one year ago
Best ResponseYou've already chosen the best response.0http://openstudy.com/study#/updates/5170a3c3e4b0aca440dfb7fb

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0$$\int_0^\infty e^{st}t^2u(t1)\, dt=\int_0^1e^{st}t^2\cdot 0\, dt+\int_1^\infty e^{st}t^2\cdot 1\, dt$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0we can split the integral into one on \((0,1)\) and another on \((1,\infty)\) because we have that the step function \(u(t1)=0\) for \(t1<0\Leftrightarrow t<1\) and \(u(t1)=1\) for \(t1>0\Leftrightarrow t>1\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the first integral is identically zero, while the second we evaluate using integration by parts: $$\begin{align*}\int_1^\infty e^{st} t^2\, dt&=\frac1s t^2e^{st}\bigg_1^\infty+\frac2s\int_1^\infty te^{st}\, dt\\&=\left[\frac1s t^2e^{st}\frac2{s^2}te^{st}\right]_1^\infty +\frac2{s^2}\int_1^\infty e^{st}\, dt\\&=\frac1se^{st}+\frac2{s^2}e^{st}+\frac2{s^2}\left[\frac1se^{st}\right]_1^\infty\\&=\left(\frac1s+\frac2{s^2}+\frac2{s^3}\right)e^{s}\end{align*}$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0which is the same thing you get after shifting \(t\to t+1\) so that \(t^2\to(t+1)^2=t^2+2t+1\) and pulling out the factor of \(e^{s}\) from \(e^{st}\to e^{s(t+1)}=e^{s}e^{st}\), so: $$\mathcal{L}\{t^2 u(t1)\}=e^{s}\mathcal{L}\{t^2+2t+1\}=\left(\frac1s+\frac2{s^2}+\frac2{s^3}\right)e^{s}$$ considering that $$\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}$$

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.0\(F(p) =\mathcal {L} \{ t^2 \; \; u(t1)\} \) [\(\bar t = t 1\)] \( F(p) = \mathcal {L} \; \left\{(\bar t+1)^2 \; u(\bar t) \right\}\) \(= \mathcal {L} \; \left\{ ( \bar t^2 + 2 \bar t + 1 \; )u(\bar t) \right\}\) \(= \left( \dfrac{2}{p^3} + \dfrac{2}{p^2} + \dfrac{1}{p} \right) e^{p}\) dw:1443657391199:dw
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