# Tagged Questions

The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

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### Inverse Laplace Transform of $\frac{1}{s} - \frac{a}{s^2 + a s^{3/2} \coth\sqrt{s}}$

I got a problem for inverse Laplace transform when solving a PDE, the solution in Laplace space is $$\widehat{f}(s) = \frac{1}{s} - \frac{a}{s^2 + a s^{3/2}\coth{\sqrt{s}}}$$ where $a$ is a ...
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### Second order differential equation with Heaviside function

I have a differential equation of the form $$y''(x) - a y(x) + b \theta(c - x) = 0, \quad y(0) = 0, \quad \lim_{x \to \infty} y(x) = 0,$$ where $a$, $b$, $c$ are some constants and $\theta(с - x)$ is ...
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### On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right)$

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
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### Solving $xy'+y=x^{k}$

Find a solution to: $$xy'+y=x^{k}$$ Where $k>0$, and on the assumption that the transforms of $f$ and $f'$ exist. I understand that we can take the Laplace of all of the terms and then find ...
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### Calculate the Laplace Transform :

Show that, provided a>0 and f is a real function that : $L\left[ f\left( t-a\right) H\left( t-a\right) \right] =e^{-pa}L\left( f\left( t\right) \right)$ I understand that when we multiply a ...
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### Convolution of $te^{2t}$ and $\delta_1-\delta_2$?

I seek to find $f*g$ where $f=te^{2t}$ and $g=\delta_1-\delta_2$ and $\delta_a(t)= \displaystyle \lim_{\epsilon \to 0^+}d_{a,\epsilon}(t)$; i.e. $\delta$ is the Dirac Delta function. We have learned ...
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### A Function of a Convolution (Laplace)

A paper I am reading makes the following claim: Assume that $a_n$ is a series of of positive, distinct, real numbers. Assume that $\epsilon_n$ are independent random standard exponential variables. ...
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### Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
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### Inverse Laplace transform of $\sqrt{H(s)}$

In there any way to find inverse Laplace transform of a function in the following general form F(s)=\sqrt{\dfrac{a_n s^n+a_{n-1}s^{n-1}+\cdots+a_1 s+a_0}{b_n s^n+b_{n-1}s^{n-1}+\...
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### Solve transport equations by using Laplace transform

I'm trying to solve rather formally one-dimensional transport equation: $$u_{t}+cu_{x}=0\quad\text{in (0,\infty)\times(-\infty,\infty)}$$ with an initial data $u_{0}$, which is bounded and ...
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### Laplace transform of a signal?

Finding the Laplace transform of a signal: How do you setup the step function $f(t)$ (equation of the graph on the image). Even though, I do know know how they setup the equations. I do know how to ...
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### Laplace transform of $\sin(\sqrt t)$

How can I use this differential equation $$4tf''(t) +2 f'(t) + a^2 f(t)=0$$ to show that $$L(\sin(\sqrt{t}))=\frac{1}{2}\sqrt{\pi}\,\frac{1}{s^{\frac{3}{2}}}\,e^{\frac{-1}{4s}}$$
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### Difficult Inverse Laplace Transform

I've had this question in my exam, which most of my batch mates couldn't solve it.The question by the way is the Laplace Transform inverse of $$\frac{\ln s}{(s+1)^2}$$ A Hint was also given, which ...
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### Finding the inverse Laplace of this function

Finding the inverse Laplace transform of $L^{-1}\left( \dfrac {1}{\left( x+1\right) ^{2}}\right)$ I understnd that the inverse laplace of $L^{-1}\left(\dfrac {1}{\left( x+1\right) }\right)$ Is ...