Laplace Transform of ${t}^{2}$ , for$t\ge 1$

$$f(t)={t}^{2},t>=1$$

what is the laplace transform of f(t)?

$$f(t)={t}^{2},t>=1$$

what is the laplace transform of f(t)?

fluerkg
2022-10-21
Answered

Laplace Transform of ${t}^{2}$ , for$t\ge 1$

$$f(t)={t}^{2},t>=1$$

what is the laplace transform of f(t)?

$$f(t)={t}^{2},t>=1$$

what is the laplace transform of f(t)?

You can still ask an expert for help

Miah Scott

Answered 2022-10-22
Author has **19** answers

$$f(t)={t}^{2}H(t-1)=[(t-1{)}^{2}+2(t-1)+1]H(t-1)$$

$$\mathcal{L}\{f(t)\}=\mathcal{L}\{[(t-1{)}^{2}+2(t-1)+1]H(t-1)\}={\mathrm{e}}^{-s}[\frac{2}{{s}^{3}}+2\cdot \frac{1}{{s}^{2}}+\frac{1}{s}]$$

$$\mathcal{L}\{f(t)\}=\mathcal{L}\{[(t-1{)}^{2}+2(t-1)+1]H(t-1)\}={\mathrm{e}}^{-s}[\frac{2}{{s}^{3}}+2\cdot \frac{1}{{s}^{2}}+\frac{1}{s}]$$

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Find inverse Laplace transform

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Find the Laplace transforms of the following time functions.

Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables.

a)$f(t)=1+2t$
b)$f(t)=\mathrm{sin}\omega t\text{Hint: Use Euler\u2019s relationship,}\mathrm{sin}\omega t=\frac{{e}^{(}j\omega t)-{e}^{(}-j\omega t)}{2j}$

c)$f(t)=\mathrm{sin}(2t)+2\mathrm{cos}(2t)+{e}^{-t}\mathrm{sin}(2t)$

Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables.

a)

c)

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find the given inverse Laplace transform

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Determine the values of r or which the given differential equation has solutions of the form $y={e}^{rt},\text{}y{}^{\u2033}-3y{}^{\u2033}+2{y}^{\prime}=0$