Chapter 7: Q9RP (page 415)

In Problems 3-10, determine the Laplace transform of the given function. $t^{2} u (t - 4)$

Short Answer

Expert verified

Therefore, the solution is. $L {t^{2} u(t-4)} {=e}^{-4s} (\frac{2}{s^{3}} + \frac{8}{s^{2}} + \frac{16}{s})$

Step by step solution

Given Information

The given function value is $t^{2} u (t - 4)$

Determining the Laplace transform

Using the following Laplace transform property, to find the Laplace of given integral:

$L {f (t) u (t-a)} {=e}^{at} L {f (t+a)}$

Let. $f (t) = t^{2}$ On applying linearity of Laplace and we get:

$\begin{matrix} L {t^{2} u(t-4)} {=e}^{-4s} L{f(t+4)} \\ {=e}^{-4s} L {{(t+4)}^{2}} \\ {=e}^{-4s} L {t^{2} +8t+16} \\ {=e}^{4s} (L {t^{2}} +8L {t} +16L{1}) \end{matrix}$

Simplify further as follows:

$\begin{matrix} L {t^{2} u(t-4)} {=e}^{-4s} (\frac{2!}{s^{3}} +8 \frac{1!}{s^{2}} +16 \frac{1}{s}) \\ {=e}^{-4s} (\frac{2}{s^{3}} + \frac{8}{s^{2}} + \frac{16}{s}) \end{matrix}$

Therefore $L {t^{2} u(t-4)} {=e}^{-4s} (\frac{2}{s^{3}} + \frac{8}{s^{2}} + \frac{16}{s})$

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In Problems 3-10, determine the Laplace transform of the given function. $t^{2} u (t - 4)$

Short Answer

Step by step solution

Given Information

Determining the Laplace transform

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In Problems 3-10, determine the Laplace transform of the given function.t2u(t-4)

Short Answer

Step by step solution

Given Information

Determining the Laplace transform

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 3-10, determine the Laplace transform of the given function. $t^{2} u (t - 4)$