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

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

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

ChmE
 2 years ago
Best ResponseYou've already chosen the best response.0Your work is cut off pretty badly

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

Miyuru
 2 years ago
Best ResponseYou've already chosen the best response.0I can't see the rest of the steps @UnkleRhaukus how can i see that. ?

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

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.1now what can we do

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3thst the final answer, what u got

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3though 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

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

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.1apply that to the denominator ,?

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3you entire answer is fine

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3i just suggested another way to simplify the numerator from \( (p+3)^3(p3)^3\)

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.1i not sure where you mean

hartnn
 2 years ago
Best 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 ?

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.1oh you mean like that

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

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.1your probably right

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3\((p+3)^3 = (p+3)(p^23p+9)\)

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3\((p3)^3=(p3)(p^2+3p+9)\)

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3rest of the solution is correct welcome ^_^
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