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Find the Laplace transform of the following function
\[g(t)=t^2\sinh(3t)\]
 one year ago
 one year ago
Find the Laplace transform of the following function \[g(t)=t^2\sinh(3t)\]
 one year ago
 one year ago

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UnkleRhaukusBest ResponseYou've already chosen the best response.1
\[\begin{align*} \\\text{(b)}&&g(t)=t^2\sinh(3t)\\ \\&&G(p)=\mathcal L\{t^2\sinh(3t)\}\\ \\&&=\int\limits_0^\infty t^2\sinh(3t)e^{pt}\text dt\\ \\&&=\int\limits_0^\infty t^2\left(\frac{e^{3t}e^{3t}}{2}\right)e^{pt}\text dt\\ \\&&=\frac12\int\limits_0^\infty t^2\left(e^{3t}e^{3t}\right)e^{pt}\text dt\\ \\&&=\frac12\left[\int\limits_0^\infty t^2e^{(p3)}\text dx\int\limits_0^\infty t^2e^{(p+3)}\text dt\right]\\ \\&&=\frac12\left[\frac{2!}{(p3)^3}\frac{2!}{(p+3)^3}\right]\\ \\&&=\frac{1}{(p3)^3}\frac{1}{(p+3)^3}\\ \\&&=\frac{(p+3)^3(p3)^3}{(p3)^3(p+3)^3}\\ \\&&=\frac{(p+3)(p^2+6p+9)(p3)(p^26p9)}{\left((p3)(p+3)\right)^3}\\ \\&&=\frac{(p^3+6p^2+9p+3p^2+18p+27)(p^36p^29p3p^2+18p+27)}{\left(p^29\right)^3}\\ \\&&=\frac{18p^2+18p}{\left(p^29\right)^3}\\ \\&&=\frac{18p(p+1)}{\left(p^29\right)^3}\\ \end{align*}\]
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
or should i leave my solution as \[G(p)=\frac{1}{(p3)^3}\frac{1}{(p+3)^3}\]
 one year ago

ChmEBest ResponseYou've already chosen the best response.0
Your work is cut off pretty badly
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
don't leave that answer the difference form, what simplification u have done was required, i think...
 one year ago

MiyuruBest ResponseYou've already chosen the best response.0
I can't see the rest of the steps @UnkleRhaukus how can i see that. ?
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
\(\begin{align*} \\\text{(b)} &g(t)=t^2\sinh(3t)\\ \\ &G(p)=\mathcal L\{t^2\sinh(3t)\}\\ \\ &=\int\limits_0^\infty t^2\sinh(3t)e^{pt}\text dt\\ \\ &=\int\limits_0^\infty t^2\left(\frac{e^{3t}e^{3t}}{2}\right)e^{pt}\text dt\\ \\ &=\frac12\int\limits_0^\infty t^2\left(e^{3t}e^{3t}\right)e^{pt}\text dt\\ \\ &=\frac12\left[\int\limits_0^\infty t^2e^{(p3)}\text dx\int\limits_0^\infty t^2e^{(p+3)}\text dt\right]\\ \\ &=\frac12\left[\frac{2!}{(p3)^3}\frac{2!}{(p+3)^3}\right]\\ \\ &=\frac{1}{(p3)^3}\frac{1}{(p+3)^3}\\ \\ &=\frac{(p+3)^3(p3)^3}{(p3)^3(p+3)^3}\\ \\ &=\frac{(p+3)(p^2+6p+9)(p3)(p^26p9)}{\left((p3)(p+3)\right)^3}\\ \\ &=\frac{(p^3+6p^2+9p+3p^2+18p+27)(p^36p^29p3p^2+18p+27)}{\left(p^29\right)^3}\\ \\ &=\frac{18p^2+18p}{\left(p^29\right)^3}\\ \\ &=\frac{18p(p+1)}{\left(p^29\right)^3}\\ \end{align*}\)
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
now what can we do
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
thst the final answer, what u got
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
though i may suggest another way to \((p+3)^3(p3)^3\) using a^3b^3 formula would simplify it easier..... rather than (a+b)^3 twice..but not much difference
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
what formula do you mean \[a^3b^3=(ab)(a^2+ab+b^2)\]?
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
apply that to the denominator ,?
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
you entire answer is fine
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
i just suggested another way to simplify the numerator from \( (p+3)^3(p3)^3\)
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
i not sure where you mean
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
\((p+3)^3(p3)^3 \\ (p+3p+3)(p^2+6p+9+p^29+p^26p+9) \\ (6)(3p^2+9)\) did i make any error ?
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
oh you mean like that
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
@UnkleRhaukus how di u get\( 18p^2+18p\) i am getting 18p^2+54
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.1
your probably right
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
\((p+3)^3 = (p+3)(p^23p+9)\)
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
\((p3)^3=(p3)(p^2+3p+9)\)
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
rest of the solution is correct welcome ^_^
 one year ago
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