Is it possible to compute the inverse Laplace transform of:

$$\frac{1}{1-{e}^{-sa}}$$

where a>0 ?

$$\frac{1}{1-{e}^{-sa}}$$

where a>0 ?

Iris Vaughn
2022-10-11
Answered

Is it possible to compute the inverse Laplace transform of:

$$\frac{1}{1-{e}^{-sa}}$$

where a>0 ?

$$\frac{1}{1-{e}^{-sa}}$$

where a>0 ?

You can still ask an expert for help

canhaulatlt

Answered 2022-10-12
Author has **17** answers

A possible solution is as follows.

$${(1-{\mathrm{e}}^{-sa})}^{-1}=\sum _{n=0}^{\mathrm{\infty}}{\left({\mathrm{e}}^{-sa}\right)}^{n}$$

Now, the inverse laplace transform of ${e}^{-nsa}$ is $Dirac(t-an)$

Then we have

$$\sum _{n=0}^{\mathrm{\infty}}\mathit{D}\mathit{i}\mathit{r}\mathit{a}\mathit{c}(t-an)$$

$${(1-{\mathrm{e}}^{-sa})}^{-1}=\sum _{n=0}^{\mathrm{\infty}}{\left({\mathrm{e}}^{-sa}\right)}^{n}$$

Now, the inverse laplace transform of ${e}^{-nsa}$ is $Dirac(t-an)$

Then we have

$$\sum _{n=0}^{\mathrm{\infty}}\mathit{D}\mathit{i}\mathit{r}\mathit{a}\mathit{c}(t-an)$$

asked 2022-10-26

Find the Laplace transform of $f(t)=1+(1-t){u}_{1}(t)+(t-2){u}_{3}(t)$

asked 2022-10-16

Given the initial problem:

$$\ddot{x}+4x=f(t),x(t=0)=3,\dot{x}(t=0)=-1$$

So I started:

$${s}^{2}(X(s)-sx(0)-\dot{x}(0)+4X(s)=\mathcal{L}(f(t))$$

Now substitute the given values:

$${s}^{2}X(s)-3s-(-1)+4X(s)=\mathcal{L}(f(t))$$

Rearranging:

$$X(s)({s}^{2}+4)-3s+1=\mathcal{L}(f(t))$$

The answer give: The Laplace transform is of the form:

$$X(t)=A\mathrm{cos}2t+B\mathrm{sin}2t+\frac{1}{2}{\int}_{0}^{t}f(\tau )\mathrm{sin}2(t-\tau )d\tau $$

Is there anybody that can help me to get the given form?

$$\ddot{x}+4x=f(t),x(t=0)=3,\dot{x}(t=0)=-1$$

So I started:

$${s}^{2}(X(s)-sx(0)-\dot{x}(0)+4X(s)=\mathcal{L}(f(t))$$

Now substitute the given values:

$${s}^{2}X(s)-3s-(-1)+4X(s)=\mathcal{L}(f(t))$$

Rearranging:

$$X(s)({s}^{2}+4)-3s+1=\mathcal{L}(f(t))$$

The answer give: The Laplace transform is of the form:

$$X(t)=A\mathrm{cos}2t+B\mathrm{sin}2t+\frac{1}{2}{\int}_{0}^{t}f(\tau )\mathrm{sin}2(t-\tau )d\tau $$

Is there anybody that can help me to get the given form?

asked 2022-11-05

How to work out the Laplace transform with respect to t of:

$$\mathrm{sin}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}$$

I know that the transform of $\mathrm{sin}\left(t\right)$ is $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{a}{{s}^{2}+{a}^{2}}}$, and transform of $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}\phantom{\rule{thinmathspace}{0ex}}$ is ${s}^{2}F\left(s\right)-s\phantom{\rule{thinmathspace}{0ex}}f\left(0\right)-s\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime}\left(0\right)$

$$\mathrm{sin}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}$$

I know that the transform of $\mathrm{sin}\left(t\right)$ is $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{a}{{s}^{2}+{a}^{2}}}$, and transform of $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}\phantom{\rule{thinmathspace}{0ex}}$ is ${s}^{2}F\left(s\right)-s\phantom{\rule{thinmathspace}{0ex}}f\left(0\right)-s\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime}\left(0\right)$

asked 2021-02-20

Find the inverse Laplace transform of (any two)

i)$\frac{({s}^{2}+3)}{s({s}^{2}+9)}$

ii)$\mathrm{log}\left(\frac{(s+1)}{(s-1)}\right)$

i)

ii)

asked 2022-01-20

Solve fractional differential equation?

$\frac{{d}^{2}}{{dx}^{2}}u\left(x\right)+b\frac{{d}^{\frac{1}{k}}}{{dx}^{\frac{1}{k}}}u\left(x\right)+cu\left(x\right)=0$

assuming$(a,b,c)=const$ and k a parameter?

assuming

asked 2021-05-05

Find f(t).

${L}^{-1}\left\{\frac{{e}^{-\pi s}}{{s}^{2}+1}\right\}$

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A-stability of Heun method for ODEs

I'm trying to determine the stability region of the Heun method for ODEs by using the equation$y\prime =ky$ , where k is a complex number.

I'm trying to determine the stability region of the Heun method for ODEs by using the equation