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 transform of composition of a step/sign function over derivative

Following this question, I was wondering if there is an analytical solution (i.e., closed form) for the composition of a step/sign function over derivative of a time-variant variable. Consider $$\...
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Calculating the inverse Laplace transform of $F_{5}(s)=\frac{-A_{10} e^{-A_{7} \sqrt s }}{(\sqrt s+A_8) \sqrt{S+\theta_c}(S-A_4)}$ by residue theorem

I have calculated the ILT by residue theorem, but the result is wrong. Would you like to check the calculation. Thanks.
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Differentiation of Laplace transforms

I have the following differential equation $ty'+(4t+1)y=-7$ with the condition $y(0)=-7$ I need to calcualte the derivative $\frac{dY(s)}{ds}$ So far I am taking the laplace transforms $$\mathscr{L}(...
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Nonlinear quasi-exponential functions

I continue the study of gradient differential equations, where bell-shaped functions are used as a function, for example: $\frac{dx}{dt}=\frac{d}{dx}(sech(x)^2)$ The solution of which is: $x(t) = \...
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Laplace transform of the integral of the solution [closed]

I have the following differential equation $y'+4y=11t^7$ with the initial condition $y(0)=3$ I need to find the following laplace transform $$L\left\{\int_0^t y(\mathcal{T}) d\mathcal{T}\right\}$$ I'...
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Second Shifting Theorem

I need to find the Laplace transform of the following function $\frac{1}{4} tu(t-7)$ using the second shifting theorem My working is as follows $\frac{1}{4} tu(t-7) = e^{-7s} L(\frac{1}{4}(t+7))$ $= ...
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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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solve $t^{2}u''+tu'+(t^{2}-1)u=0$ using Laplace

given the ode:$t^{2}u''+tu'+(t^{2}-1)u=0;u(0)=1,u'(0)=0$ solve with Laplace transform. My try: $t^{2}u''+tu'+(t^{2}-1)u=0\\Lu''=s^{2}Lu-1\\Lu'=sLu-1\\t^{2}(s^{2}Lu-1)+t(sLu-1)+(t^{2}-1)Lu=0\\t^{2}s^{2}...
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Solving IVP via Laplace transform

I have the following initial value problem (IVP): $$ \mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}, \qquad \mathbf{X}(0)=\mathbf{I} $$ $$ \text { If } \mathbf{x}(s)=\mathscr{L}\{\mathbf{X}(t)\}=\mathscr{L}...
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Second order linear ODE equal to pulse funcion with Laplace transform method

The question is to solve the following initial conditions problem with the Laplace transform method. $$ f'' + 2f' -3f = \begin{cases} 1, \ 0 \leq t < c \\ 0, \ t \geq c \end{cases}; f(0) = f'(0) = ...
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Laplace's equation and a complex integral

In Schrödinger's first paper on wave mechanics, he mentioned a general type of "Laplace's equation": $$U''+\left(\delta_0+\frac{\delta_1}{r}\right)U'+\left(\epsilon_0+\frac{\epsilon_1}{r}\...
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solution verification: $y'''-3y'+2y=24e^{t}$ with laplace

I tried to solve this ode, but the solution seems to be very strange. solve: $y'''-3y'+2y=24e^{t}$ given that $y''(0)=y'(0)=y(0)=0$. $s^{3}Ly-3sLy+2Ly=\frac{24}{s-1}\\Ly(s^{3}-3s+2)=\frac{24}{s-1}\\Ly=...
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A formula for $\int_0^\infty \frac{t^k}{\Gamma(t)}dt$: generalized Fransén-Robinson constants?

How to prove the following formula, providing a generalization of the Fransén-Robinson constant? $$\boxed{ \int_0^\infty \frac{t^k}{\Gamma(t)} dt = \sum_{n=1}^\infty \frac{n^{k+1}}{n!} + (2k)!!\sum_{...
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solve $y''-3y'+2y=g$ whereas $g(t)=\begin{cases} 0, & t>\pi \\ \sin t,& 0\leq t\leq\pi. \end{cases}$

Solve $$\left\{\begin{aligned} &y''-3y'+2y=g\\ &y(0)=y'(0)=0 \end{aligned}\right.$$ whereas $$g(t)=\begin{cases} 0, & t>\pi \\ \sin t,& 0\leq t\leq\pi. \end{cases}$$ My try: \...
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solve with Laplace $y''-2y'+2y=0$ detect my mistakes

$y''-2y'+2y=0$ and $y(0)=2,y'(0)=0$. My try: Ly will be the Laplace function of y. $\begin{array}{c} y''-2y'+2y=0\\ y(0)=2,y'(0)=0\\ Ly''=sLy'\\ Ly'=sLy-2\\ Ly''=s(sLy-2)=s^{2}Ly-2s\\ s^{2}Ly-2s-2(sLy-...
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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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$L^2$ norm of function from its Laplace transform

I saw in this control theory textbook (p. 39) two ways of computing the $L^2$ norm of a univariate function. In the time domain, we have $$||u(t)||_2 = \left(\int_{0}^{\infty}\lvert u(t) \rvert^2\,dt\...
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M/G/1 queue with the server serving up to $m$ customers simutaneously

Consider an M/G/1 Queue with the modification that the server may serve up to $m$ customers simultaneously. If the queue leanth if less than or equal to $m$ at the beginning of a service period then ...
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Laplace transformation using the definition and multiplication rule

I'm trying to use the multiplication rule with the definition of the Laplace transformation of: $\mathcal{L} [te^{2t}](s)$ with $Re(s)>2$. I get an error and I don't understand what I'm doing wrong:...
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Laplace transform of $f(t)/t$ derived from Laplace transform of $tf(t)$

I am trying to derive the Laplace transform of $$g(t) = \frac{f(t)}{t}$$ from knowing that the laplace transform of $tv(t)$ is $-V'(s)$ I'm doing OK, but it's the limits of integration I'm having an ...
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which types of Differential equations can be solved using Laplace transformation? which are not possible to solve using Laplace transform?

please if you are not going to answer and instead tell me that I should know this because its 3rd grade math then simply dont respond. I never get straightforward answers on this forum, only sarcastic ...
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Laplace transform of unit step with negative arguments

Im having a problem integrating $u(-t)$ so I can get the Laplace transform. My table shows $\mathcal {L}\{-u(-t)\} = 1/s$ but I'm not sure if there's a property I should be using for negative ...
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How can one implement a rational function in frequency domain using integrators in time domain?

There is a simple method to implement a rational function in frequency domain using integrators and feedback in time domain. Could somebody draw a nice picture and show which coefficient of the ...
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Laplace-Stieltjes Transform of a sum

Let $N$ be a geometrically distributed random variable with parameter $\frac{1}{2}$, i.e., $\mathbb{P}(N=n)=(\frac{1}{2})^{n+1},n\geq0$, and let $Y_{1},Y_{2},...$ be independent identically ...
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Missing constant after Laplace inverse

I took this Cauchy problem \begin{align} &y''-3y'+2y=4t+8,\\ &y(0) = 2,\\ &y'(0) =7 \end{align} And transformed it through Laplace transform $$ Y(s)=\frac{4-8s}{s^2(s-1)(s-2)} + \frac{1+2s}...
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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 ...
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1answer
71 views

Laplace Transform: Product of Bessel Functions

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 >...
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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\...
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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 ...
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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 ...
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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 ...
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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\...
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120 views

Using Laplace transform to evaluate definite integrals?

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 $\...

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