# 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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### Pde using laplace transform

Could you help me to find a solution for this partial differntial equation by using laplace transform $$u_{t} - u_{xx} = xt$$ where $$u(0,t)=t, \quad u(1,t)=t^2, \quad u(x,0)= \sin \pi x$$ I tried ...
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### Is the Laplace transform essentially a generalized version of the Fourier transform?

My current understanding of the two concepts is far from perfect, and I am essentially just a beginner. But it seems to me that while the Fourier tries to decompose functions as a composition of ...
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### inverse Laplace transform of gamma function

My problem is to get the inverse Laplace transform of the following equation. $$\hat{P}(s) = \frac{\Gamma(p+1+s T)}{p! N^{s T}}$$ $p$, $T$ and $N$ are positive constants. The denominator $N^{-s T}$ ...
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### Partial fraction decomposition of $\frac{21}{s^{2}+4}$ for inverse-Laplace transform

So I have this number which I want to do inverse-Laplace transformation on, which is kind of complicated. So it would be easier to do some partial fraction decomposition first. I am trying to do the ...
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### Solving for $x$ in a Laplace equation

So I have this Laplace equation: $$s^{2}x+4sx+5=\frac{s}{s-1}$$ And I want to solve for $x$. My result is the following: $$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$ Which is also the same answer that for ...
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### Solving Bessel's equation by Laplace transform

I am learning Bessel function the solution of Bessel equation by book 'Advanced Engineering Mathematics' by Peter V.O'Neil and here i found its derivation by Laplace transform. In this derivation of ...
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### Laplace transform identity $F(s) = \mathcal{L}(t^{-3/2} \mathrm{e}^{-1/t})$

I'm asked to prove the following result: If $F(s)$ is the Laplace transform of $f(t) = t^{-3/2} \mathrm{e}^{-1/t}$, show that $F'(s)=-s^{-1/2}F(s)$. I'm having a lot of troubles to prove this ...
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### Laplace Transform of Dirac Delta function

I've seen everywhere that that the Laplace Transform of Dirac Delta function is: $$L[\delta(t-a)] = e^{-sa} \text{ when } a > 0$$ But they never explain what happens when $a < 0$. Can I assume ...
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### Laplace Question $f(t) = e^{-t} \sin(t)$

I need help with this Laplace question. $$f(t) = e^{-t} \sin(t)$$ Answer should be $\dfrac{1}{s^2 + 2s + 2}$ What I'm currently doing is as follows: $u = \sin(t)\qquad$ ...
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### How to write a transfer function (in Laplace domain) from a set of linear differential equations?

Provided I have a system of linear differential equations (in time domain) such as: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
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### Dynamic real-time system problem

I am struggling with a systems theory problem, the task is as follows: u(t) -> H(s) -> y(t) H(s) being the transfer function $$H(s) = H(s) = \frac{s+1}{s(s+2)^{2}}$$ $$u(t) = e^{-5t}$$ So ...
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### Why M.G.F transform is injective a.s.?

We always use the theorem that If we know a random variable's MGF, we can determine its Pdf, which means the map from Pdf to Mgf is injective almost surely. And I just wanna know why this is ture.
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### What topics should I study to understand Laplace transform?

If I'm a beginner to start understanding Laplace transform, from where should I start to understand Laplace Transform?
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Let $f:[0,\infty) \rightarrow \mathbb{R}$ be continuous, with the property that $f(t)e^{-pt} \rightarrow 0$ as $t \rightarrow \infty$. If $\mathcal{L}[f(t)] = \hat f(p)$ then $\mathcal{L}[e^{at}f(t)]... 2answers 73 views ### Inverse Laplace Transform of$e^{\frac{1}{s}-s}$doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert: $$F(s) = e^{\frac{1}{s}-s}$$ I can't find it in any table and the strong singular growth ... 0answers 46 views ### Solution of partial differential equation - modified heat equation I want to solve the "modified" heat equation $$\frac{\partial y}{\partial t}=a\frac{\partial^2 y}{\partial x^2} +b\frac{\partial y}{\partial x} +cy+d$$ I assumed that a, b, c and d are all constant ... 1answer 42 views ### How to solve nonlinear partial differential equation with two variables somehow, I got this partial differential equation but I don't know how should I start. $$a\frac{\partial f(x,t)}{\partial x}\left[ \frac{\partial g(x,t)}{\partial t}+bg(x,t)\left[g(x,t)-f(x,t)+C\... 1answer 23 views ### In the context of Laplace transforms, what does the subscript in h(t) = f(t)\cdot u_3(t) signify? Problem Note: I do not need help solving this problem (yet), but I'm unsure about notation. Find the Laplace tranform of the function h(t) = e^{2(t-3)}u_3(t). Question What does the subscripted ... 1answer 65 views ### How do you find the Inverse Laplace transformation for a product of 2 functions? If$$\mathscr{L}(y)=\frac{ne^{-pt_0}}{n^2+\omega^2}\left(\frac{1}{p+n}+\frac{n}{p^2+\omega^2}-\frac{p}{p^2+\omega^2}\right)$$show that$$\bbox[yellow] {y=n\left(\frac{e^{-n(t-t_0)}}{n^2+\omega^2}+\... 1answer 36 views ### Find a Lebesgue integrable function which satisfies a convolution equation Let$f:\mathbb{R}^n \to \mathbb{R}$be a non-negative Lebesgue integrable function with integral on$\mathbb{R}^n$equals to 1. Let$\tilde{f}(x)=f(-x)$. Suppose$f$satisfies the following equation:$...
I'll give you the whole context: In solving the heat equation $u_t = ku_xx$ with bounds $u(x,0)=0, u(0,t)=0, u(l,t)=f(t)$, let $v(x,t)$ be the solution for the special case $f(t)=1$. Use the Laplace ...