# Questions tagged [laplace-transform]

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

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### Find The Laplace Transform With Details

How to solve the transforms below step by step $$\mathscr{L}^{-1} \frac{1}{(s+ \lambda)^2- \omega^2}$$ $$\mathscr{L}^{-1} \frac{a(s+2 \lambda)+b}{(s+ \lambda)^2- \omega^2}$$ I found some ...
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### How do you find the Laplace transform of $f''(\cos^2t)$? [closed]

Can someone show me how the plausible formulas for this, or at least the beginning?
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### In the integral of a function, why is it that I am able to take out a function and claim it is smaller than the integral itself?

Could you please explain what happens in the second last line, where they had the integral larger or equal to the e^-s integral of 1/t. I am confused as it seems that they just pulled the e^-s out and ...
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### Is this step function expressed as a power series wrong?

Here I have a step function expressed in the answer as a power series. Please start at Note that we can write... I think the power series is wrong as , at any x, the value is infinite. However it is ...
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### Is it possible, solely with the function $f(x) = \sum_{n>0} a_nx^n$, to obtain the function $\sum_{n>1} \frac{a_n}{n!} x^n$?

It popped in my head these functions seems fairly independent even tho the first one kinda should determine the other. for $a_n = 1$ we have $\sum x^n = \frac1{1-x}$ and $\sum \frac{x^n}{n!} = e^x$ ...
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### Determining a measure from a moment sequence?

I am considering the Stieljes moment problem (https://en.wikipedia.org/wiki/Stieltjes_moment_problem), and its solution for a special class of moment sequences derived from quantum mechanics. One is ...
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### Proof of Laplace transform (Stuck)

Well the professor has given us homework to do, the first question was fine but the second question has gotten me completely dumbfounded so I was wondering if somebody here had an idea on what to do ...
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### Find the solution of $y'''+y''-y'-4y=2-2x$, $y(1)=\frac{1}{2}, y'(1)=y''(1)=0$ by using Laplace transformation

I want to find the solution of $y'''+y''-y'-4y=2-2x\$, $y(1)=\frac{1}{2},\ y'(1)=y''(1)=0$ by using Laplace transformation. When I try to solve this exercise by appling Laplace transformation I get ...
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### Application of Laplace Transform in differential equations [closed]

I don't understand the steps within the red outline. Please give me the rules to do it. Here how is it possible to figure out the application of Laplace Transform? enter image description here
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### Laplace transforms of a differential equation

I have the following differentual equation $y'+4y=11t^7$ with the inital condition $y(0)=3$ I then have to calculate the Laplace transform of $y(t)$ I did the laplace tranform of $y(t)=11t^7$ which ...
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### About transfer functions and impulse,forced and natural response

Help me see if I got it right, let say I got a differential equation $${y}''(t)+4{y}'(t)+3y(t)=tu(t)$$ Transfer function is $$\dfrac{output}{input}$$ that is, $$H(s)=\dfrac{Y(s)}{X(s)}$$ Since I have ...
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### Property of Z transform

Is there any direct relationship exist between p times differentiation of F(z) and f(n) similar to Laplace transform where n times differentiation directly related to its time domain counterpart ? ...
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### Nontrivial solution of a fourth-order DE with variable coefficients$\frac{d^2}{dx^2}\left(f(x)\frac{d^2y(x)}{dx^2}\right)-ag(x)y(x)=0$

How to find the general closed-form solution of the following eigenvalue problem? \begin{equation} \frac{d^2}{dx^2}\left(f(x)\frac{d^2y(x)}{dx^2}\right)-ag(x)y(x)=0 \\ \text{where} \ 0< f(x),g(x)\...
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### Laplace transformation for time series

I have a set of covariates that follow an AR(1) dynamic: $Y_{t+1}= \alpha + \phi Y_t + e_{t+1}$, with $e_{t+1} \sim N(0, \Sigma)$ The (conditional?) Laplace transform for this dynamic is according to ...
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### Can somebody help me with the beginning of Laplace transform?

$ty''(t)+(t-1)y'(t)-y(t)=0$ $y(0)=5$ $y(+\infty)=0$ Can somebody tell me what $y(+\infty)=0$ represents?
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### If the Laplace transforms of two functions are equal, are the functions equal? [closed]

In other words, is the following true? $$\int_0^\infty f(t)e^{-st}\, dt=\int_0^\infty g(t)e^{-st}\, dt\implies f(t)=g(t)$$ If not, what are examples of different functions with the same Laplace ...
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### How to get the inverse Laplace transform of an expression with irrational function term by residue theorem?

$$F(s)=\frac{ e^{-A_2\sqrt{s}}}{s(\sqrt{s}+A_3)}$$ The difficulty is to deal with the term $(\sqrt{s}+A_3)$.
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### Finding the inverse Laplace transform of $\frac{s}{(s+1)^3}$ using inversion formula

I need to find the inverse Laplace transform of $$F(s) = \frac{s}{(s+1)^3}$$ using Bromwich Integral. The Bromwich contour will look something like this. Actually you can see this problem on the ...
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### Boundedness of the Laplace Transform on $(C[0, 1], \|\cdot\|_2)$.

Let $T:(C[0,1], \|\cdot\|_2)\to (C[0,1], \|\cdot\|_2)$ be defined by $$(Tf) (s)=\int_0^1e^{-sx}f(x)\mathrm dx, s\in[0,1].$$ Show that $T$ is a bounded operator. Is $T$ one-one? Is $T$ onto? Justify. I ...
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### Laplace Transform of $\cos(t)/t$

this seems like a homework problem. yes! To some extent. But really I was not getting it. I was not able to get the Laplace transform of $\cos(t)/t$. using the property of Integration in Laplace ...
I'm trying to find the closed form solution for the integral of the product of Bessel functions. Namely, $$I_{\alpha \beta} = \int_{0}^{\infty} dT e^{-2s T} J_{\alpha}(T) J_{\beta}(T)$$ where $s >... 0answers 54 views ### Inverting a Fourier-like expression by operating with a Laplacian The question that follows has to do with the effects of a turbulent atmosphere on wave propagation. The structure function,$D(\vec{r})$, which is defined as, \begin{equation} D(\vec{r}) = \left\... 1answer 94 views ### solution of initial value problem using laplace transformation If$g(t)=0$for$0\leq t<1$and$g(t)=t^2-$1 for$t\geq 1$.Then find solution of initial value problem$y''+2y'+3y=g(t)$and$y(0)=0,y'(0)=1$using Laplace transformation. What I tried: Taking ... 1answer 79 views ### Solving a non linear differential equation This is a first order differential equation: $$\frac{df_1}{dx} + \frac{(f_1)^2}{h^2} - \frac{2m}{h^2} \lambda \delta(x-pa)=-\frac{2mE_1}{h^2}$$ Where, h,$\lambda $and$E_1$are constants and and ... 0answers 67 views ### Why does the Laplace transform of$\ln(t)$exist? In my lecture script for Complex Analysis the following requirements are stated for a function to have a laplace transform: (1)$f(t)=0$for all$t < 0$(2) There exist a real$\sigma$and a ... 1answer 41 views ### How do I solve the following diffusion problem? I am asked to solve the following diffusion problem: $$u'_t-au''_{xx} = 0, \quad x>0, \ t>0,$$ $$u(0,t) = 0, \quad t>0,$$ $$u(x,0)=1-\theta(x-1), \quad x>0.$$ I expand this problem to$x\...
I would like to know if there is some relation between Laplace Transform and similar definite integrals. For instance, if I know that $$\mathcal{L}\{f(t)\}(s)=F(s),$$ have I some information about \$\...