Consider the complex function $f(s)=\frac{1}{\frac{l}{c}\sqrt{(s(s+{r}_{0})}}$ where ${r}_{0},l,c$ are positive real number and s is a complex variable. How I can obtain the inverse Laplace transformation of this function?

Jensen Mclean
2022-09-06
Answered

Consider the complex function $f(s)=\frac{1}{\frac{l}{c}\sqrt{(s(s+{r}_{0})}}$ where ${r}_{0},l,c$ are positive real number and s is a complex variable. How I can obtain the inverse Laplace transformation of this function?

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Jordan Owen

Answered 2022-09-07
Author has **7** answers

I have found the following formula about my question.

$${L}^{-1}\{\frac{1}{\sqrt{s+a}\sqrt{s+b}}\}={e}^{-\frac{(a+b)t}{2}}{I}_{0}(\frac{a-b}{2}t)$$

where ${I}_{0}(x)$ is modified Bessel function.

$${L}^{-1}\{\frac{1}{\sqrt{s+a}\sqrt{s+b}}\}={e}^{-\frac{(a+b)t}{2}}{I}_{0}(\frac{a-b}{2}t)$$

where ${I}_{0}(x)$ is modified Bessel function.

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find the inverse of Laplace transform

$\frac{3}{(s+2{)}^{2}}-\frac{2s+6}{({s}^{2}+4)}$

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How do I find the inverse Laplace transform of $\frac{4s}{{s}^{2}+4}}^{2$ ?

asked 2021-02-14

Use Laplace transforms to solve the following initial value problem

$y"-{y}^{\prime}-6y=0$

$y(0)=1$

${y}^{\prime}(0)=-1$

asked 2021-12-31

When a body is removed from an oven, its temperature is measured at ${230}^{\circ}F$ . Four minutes later its temperature is ${120}^{\circ}F$ . If the room temperature is found to be ${70}^{\circ}F$ , how long will it take for the chicken to cool off to a temperature of ${90}^{\circ}F$ . What is the steady state temperature of the body?

asked 2021-09-15

Consider the following initial value problem:

$y\prime \prime -4y\prime -45y=\mathrm{sin}\left(7t\right),y\left(0\right)=1,y\prime \left(0\right)=-4$

Using Y for the Laplace transform of y(t), i.e., Y=L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)=?

Using Y for the Laplace transform of y(t), i.e., Y=L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)=?

asked 2022-09-03

Use the Laplace transform table and the linearity of the Laplace transform to determine the following:

$L\{{t}^{4}-{t}^{2}-t+\mathrm{sin}\sqrt{2t}\}$

$L\{{t}^{4}-{t}^{2}-t+\mathrm{sin}\sqrt{2t}\}$

asked 2022-09-23

Laplace transform of $\frac{{\mathrm{sin}}^{2}(t)}{t}$

Because :

$$\mathcal{L}\left(\frac{f(t)}{t}\right)={\int}_{s}^{\infty}F(s)ds$$

and

$$\mathcal{L}({\mathrm{sin}}^{2}(t))=\frac{2}{s({s}^{2}+4)}$$

then

$$\int \frac{2}{s({s}^{2}+4)}ds=\frac{1}{2}\mathrm{ln}|s|-\frac{1}{4}\mathrm{ln}|{s}^{2}+4|$$

What should I do next?

Because :

$$\mathcal{L}\left(\frac{f(t)}{t}\right)={\int}_{s}^{\infty}F(s)ds$$

and

$$\mathcal{L}({\mathrm{sin}}^{2}(t))=\frac{2}{s({s}^{2}+4)}$$

then

$$\int \frac{2}{s({s}^{2}+4)}ds=\frac{1}{2}\mathrm{ln}|s|-\frac{1}{4}\mathrm{ln}|{s}^{2}+4|$$

What should I do next?