Question

L^(-1){1/(p-1)^5(p+2)} find inverse laplace transform​

Answer 1

Answer=

To find the transformation, compare the expression to the parent function and check to see if there is a horizontal or vertical shift, reflection about the x-axis or y-axis, and if there is a vertical stretch.

Answer 2

Answer:

The inverse laplace transform of given function is

$L {}^{ - 1} (f(p)) \\ = \frac{t {}^{4}e {}^{t} }{74} - \frac{t {}^{3} e {}^{t} }{54} + \frac{t {}^{2} e {}^{t} }{54} - \frac{te {}^{t} }{81} + \frac{e {}^{t} }{243} - \frac{e {}^{ - 2t} }{243}$

Step-by-step explanation:

The given function can be written as

$f(p) = \frac{1}{(p - 1) {}^{5} (p + 2)}$

It can be rearranged in way shown below

$f(p) = \frac{1}{3} ( \frac{(p + 2) - (p - 1)}{(p - 1) {}^{5} (p + 2)} ) \\ = \frac{1}{3} ( \frac{1}{(p - 1) {}^{5} } - \frac{1}{(p - 1) {}^{4}(p + 2) } )$

Doing the similar operation again for the second term we get

$f(p) = \frac{1}{3} ( \frac{1}{(p - 1) {}^{5} } - \frac{1}{3} ( \frac{1}{(p - 1) {}^{4} } - \frac{1}{(p - 1) {}^{3}(p + 2) } ) )$

Repeating again

$f(p) = \frac{1}{3} ( \frac{1}{(p - 1) {}^{5} } - \frac{1}{3} ( \frac{1}{(p - 1) {}^{4} } - \frac{1}{3} (\frac{1}{(p - 1) {}^{3}} - \frac{1}{(p - 1) {}^{2}(p + 2) } ) )$

Repeating again we get

$f(p) = \frac{1}{3} ( \frac{1}{(p - 1) {}^{5} } - \frac{1}{3} ( \frac{1}{(p - 1) {}^{4} } - \frac{1}{3} (\frac{1}{(p - 1) {}^{3}} - \frac{1}{3} (\frac{1}{(p - 1) {}^{2} } - \frac{1}{(p - 1)(p + 2)} ) ) \\ = \frac{1}{3} ( \frac{1}{(p - 1) {}^{5} } - \frac{1}{3} ( \frac{1}{(p - 1) {}^{4} } - \frac{1}{3} (\frac{1}{(p - 1) {}^{3}} - \frac{1}{3} (\frac{1}{(p - 1) {}^{2} } - \frac{1}{3}( \frac{1}{(p - 1)} - \frac{1}{p + 2} ) )$Simplifying the above term we get,

$f(p) = \frac{1}{3(p - 1) {}^{5} } - \frac{1}{9(p - 1) {}^{4} } + \frac{1}{27(p - 1) {}^{3} } - \frac{1}{81(p - 1) {}^{2} } + \frac{1}{243(p - 1)} - \frac{1}{243(p + 2)}$

Applying inverse laplace to the above function we get

$L {}^{ - 1} (f(p)) \\ = \frac{t {}^{4}e {}^{t} }{74} - \frac{t {}^{3} e {}^{t} }{54} + \frac{t {}^{2} e {}^{t} }{54} - \frac{te {}^{t} }{81} + \frac{e {}^{t} }{243} - \frac{e {}^{ - 2t} }{243}$

Therefore,the inverse laplace transform of the given function is derived.

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