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

learn more… | top users | synonyms

0
votes
1answer
45 views

Contradiction between Fourier and Laplace transforms?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has both Fourier and Laplace transforms. Also let $f(t)=0$ for all $t<0$. The Fourier transform of $f$ is ...
1
vote
0answers
21 views

Inverse Laplace transform with an arbitrary parameter

I am trying to find this: \begin{equation} \mathcal{L}^{-1}(s^nF(s))=?, \end{equation} where the parameter $n$ can be an arbitrary value. I know when $n$ is a positive integer, it can be written as a ...
0
votes
0answers
20 views

Prove that L{t*y'(t)}(s) = Y(s) - s*Y'(s) and show that L{t*y(t)}(s) = -d/ds(Y'(s))

I believe that both these questions should be solved using integration by parts and definitions of the Laplace transform. Prove that$$\displaystyle{\mathcal{L} \{ t*y'(t) \}(s) = Y(s) - s*Y'(s)}$$ ...
0
votes
3answers
24 views

How do I find the laplace transform of a product? [closed]

How do I find the laplace transform of a product? Specifically $e^{5t}\cos{t}$?
0
votes
1answer
18 views

Differentiating Laplace transform of random variable

Let $Y$ be a random variable and $A$ an event, such that $g(q):= E(e^{-qY} ; A)$ exists for all $q \geq 0$. (Here $E(X ; A) := \int_{A} X \hspace{3pt} dP$ for a random variable $X$). I want to check ...
1
vote
1answer
25 views

Use Laplace Transformations to solve $y''+2y'+5y=3e^{-x}sin(x)$, with $y(0)=0$, $y'(0)=3$

I've gotten this far and I cannot proceed: $L[y]=\frac{L[3e^{-x}sin(x)]+3}{p^2+2p+5}= \frac{3}{((p+1)^2+1)(p^2+2p+5)}+\frac{3}{p^2+2p+5}$ I'm finding it impossible to find the inverse to solve for ...
5
votes
1answer
47 views

What am I doing wrong when I try to deduce the Laplace transform formula?

The Laplace transform of a function $f(t)$ is the projection of $f(t)$ vector (indexed with $t$) onto the linearly independent set of vectors $e^{st}$. The projection of a vector $\vec{v}$ onto ...
0
votes
1answer
42 views

Solve the PDE $\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=x$ using Laplace transform in $t$

Using Laplace transform in $t$, or otherwise, solve the equation for $u$: $$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=x$$ in the region: $x$ > 0, $t$ > 0, subject to the boundary ...
0
votes
0answers
22 views

inverse laplace of {5e^(-ts)}/{s+5} with a unit step input and how to matlab plot?

I multiplied the equation with a unit step to give {5e^(-ts)}/{s(s+5)}. then I did a PFA on {5}/{s(s+5)} to get (1/s)-(1/(s+5))e^(-ts) I then used the inverse laplace transform to get ...
0
votes
1answer
33 views

laplace transform of $t^nf(t)$

I have: $$\mathcal{L}(t^nf(t)) = \int_0^\infty t^nf(t)e^{-st}\ dt = \left(-\dfrac{d}{ds}\right)^n \int_0^\infty f(t) e^{-st}$$ I don't understand where the derivative came from
0
votes
0answers
34 views

Finding a function given as a part of a convolution integral

I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \alpha} \right)} = \int_0^t \frac{f\left(x, s\right)}{t - s}ds$$ where $\alpha$ and ...
1
vote
1answer
50 views

(Laplace Method) $y'' - 4y' = 6e^{3t} - 3e^{-t}$

For this problem $y(0) = 1$ and $y'(0) = -1$ I need to solve this problem using this: \begin{align*} y(t) &\longrightarrow Y(s)\\ y'(t) &\longrightarrow sY(s) - y(0)\\ y''(t) ...
1
vote
2answers
22 views

Which rules are used to make function like one in Laplace Transformations table?

I have function like this: $$\frac{s^2+3s+3}{(2s^2+7s+7)} $$ It needs to be brought to the level of Laplace Transformations from table, like these two: $$\frac{a}{(s-b)^2 + a^2} $$ ...
0
votes
1answer
81 views

Inverse Laplace Transform With Dead Time

Here is the transfer function that needs to be transformed back into the time domain: $$Y(s)=\frac{K_{2}e^{-\theta s}}{s(\tau_{1} s + 1)(\tau_{2} s + 1)}$$ Then would the response be: $$y(t) = ...
0
votes
1answer
68 views

Non-Linear Regression for Parameter Estimation

I have a second order system, it's response to a step change can be expressed in the s-space as: $$Y(s)=\frac{K_{2}e^{-\theta s}}{s(\tau_{1} s + 1)(\tau_{2} s + 1)}$$ Which can be inverse ...
1
vote
1answer
21 views

Solving initial value problem with delta function

Use the Laplace transform to solve the following initial value problem: $$y''+4y = 3\delta(t-\pi), \quad y(0)=0, y'(0)=0$$ I solve the equation based on what I learned in class, my answer is ...
0
votes
2answers
37 views

Show that these two differential equations have the same solution

Question: Show that the problems $ax'' + bx' + cx = f(t); x(0) = 0, x'(0) = v_0$ and $ax'' + bx' + cx = f(t) + av_0 \delta(t); x(0) = x'(0) = 0$ have the same solution for $t \gt 0$. Thus the effect ...
1
vote
1answer
24 views

Signal whose Laplace transform contains derived Dirac-deltas: How do I find the inverse transform?

I must reconstruct the input signal to a system, knowing the output signal and the system transfer function. At the end, I found that the Laplace-Transform of the input signal is something like: $$ ...
2
votes
2answers
57 views

Solving convolution problem with $\delta(x)$ function

Suppose we had the functions: $$g(t)=\theta(t)(e^{-t}+2e^{-2t})+2\delta(t)$$ and $$u(t)=2(\theta(t)-\theta(t-2))$$ Then we have ...
0
votes
0answers
14 views

Can this be expressed in terms of Laplace Transform?

I have an expression in the from: $$\mathbb{P}(H>\theta)=\int_0^{\infty}\exp(-m\theta I)f_I(i){\rm{d}}i$$ Here, $f_I(i)$ is the PDF of random variable $I$. Let, $\mathcal{L}_I(s)$ is the Laplace ...
1
vote
2answers
54 views

IVP solved with Laplace transformation - mistake?

I want to solve following IVP with Laplace transformation. \begin{align} x''(t) + 2x'(t) + x(t) &= \begin{cases} 0, & t < 0\\ 1, & t \in (0;2)\\ 3, & t > 2 \end{cases}\\ x(0) ...
0
votes
1answer
28 views

Find the Laplace transform f(s) for the figure showing below

Figure I tried to solve it and came up with this solution.
0
votes
2answers
17 views

Definition of Laplace

If a function is not of exponential order, then is it possible that function's Laplace transform exist? If yes, then how can we determine for a function that its Laplace transform exists or not?
0
votes
0answers
28 views

How to solve the laplace transform of $f_m(t_m)$ = $f_1(t)$ $\int_{0}^{\alpha} f_2(\tau) d\tau$ + $f_2(t)$ $\int_{0}^{\alpha} f_1(\tau) d\tau$.

Could you please help me to solve the following : if $t_m$ = min($t_1$,$t_2$) The probability density function $t_1$ is $f_1(t_1)$ and $t_2$ is$f_2(t_2)$ then $f_m(t_m)$ is the probability density ...
1
vote
1answer
16 views

Having some trouble with inverse Laplace tranform

How to solve this using inverse Laplace transform? 1/[($s$+1)($s$+2)$^4$] I though of this solution which is $A$/($s$+1) + $B$/($s$+2) + $C$/($s$+2)$^2$ + $D$/($s$+2)$^3$ + $E$/($s$+2)$^4$ Then ...
0
votes
0answers
30 views

Inverse Laplace Transform of an Infinite Sum

How to find the Inverse Laplace Transform of the following expression $$1+\frac{-Xs^{2/a}-Ys^{3/b}}{1!}+\frac{(-Xs^{2/a}-Ys^{3/b})^2}{2!}+\cdots$$ Any approximation is also okay... Here $a$ and $b$ ...
1
vote
1answer
42 views

Inverse Laplace Transform of Stretched Exponential

I have a Laplace tranform in the form given below $$\mathcal{L}(s)=\text{exp}(-As^{2/\alpha}-Bs^{3/\beta})$$ which is an multiplication of two stretched exponential decay function where $A,B >0$ ...
1
vote
1answer
39 views

Unilateral Laplace Transform vs Bilateral Fourier Transform

I would like to know why when we find the Laplace transform we use the one-sided (unilateral) version (all Laplace transform tables I can find are one-sided, like this one ...
0
votes
0answers
20 views

How to obtain a stabilization problem in linear system with controller?

The scheme of system: The equasion after Laplace transform: $$Y(p) = \frac{PID(p)\cdot H(p)}{1 + PID(p)\cdot H(p)} Y^d(p)$$ Now I want to make inverse Laplace transform and then plot $y(t)$, but ...
-1
votes
2answers
48 views

How to determine if $z(x,y)=\ln(x^2 + y^2)$ is a harmonic function [closed]

So I know that a function is harmonic if it satisfies Laplace's equation: $\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = 0$ But I'm just not sure how I should put it in ...
0
votes
2answers
26 views

Find a solution to $xy'+y = x^k$ using the Laplace transform.

The part that is giving me trouble is: $$\mathcal{L}[xy']$$ I have never done this before so I ploughed through. $$\mathcal{L}[xy'] = \int^{\infty}_0ty'(t)e^{-pt}dt = ...
0
votes
1answer
18 views

Having trouble finding inverse Laplace Transform

I have this Laplace transform $$X(s)=\frac{1}{s\cdot(s^{2}+0.2s+1)}$$ I want to find its inverse transform. I did the following. First I decomposed that into partial fractions ...
1
vote
1answer
22 views

Laplace and unit step- multiplication vs convolution

Please be gentle if the question is stupid. When using the laplace transform, you often multiply the function of interest by a shifted unit step function to operate on the positive portion of the ...
0
votes
0answers
33 views

Prove this inequality using laplace transform

Let $n$ be a strictly positive integer, and $a_1,\cdots a_n,b_1,b_2 \cdots b_n$ strictly positive real numbers. Prove that $$\sum_{i=1}^n (\frac{a_i}{b_i})^2 \le 2\sum_{1\le j,i \le n} ...
2
votes
1answer
50 views

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 ...
3
votes
2answers
58 views

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 ...
2
votes
3answers
52 views

Find f(x) $\int_0^x f(u)du - f'(x) = x$

Find f(x) $$\int_0^x f(u)du - f'(x) = x$$ I was not given f(0) which makes it difficult for me to find f(x). This is what I have thus far: $$\frac{F(p)}{p}-pF(p)+f(0)=\frac{1}{p^2}$$ ...
1
vote
1answer
38 views

Laplace Transform: $g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u) du$ [closed]

$$g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u)du$$ I need to find $g(x)$ I believe I need to use Laplace Transform with this in mind (Convolution Thm): $$(f*g)(x)= \int_0^x f(x-t)g(t)dt$$ However I don't ...
0
votes
1answer
47 views

Two approaches to problem give different answers — which one is correct?

I approached this problem in two ways and arrived at different answers. Both ways seem logical to me. Are they both correct, or is one flawed? This is the original problem: $$\mathfrak L^{-1} ...
2
votes
2answers
58 views

Laplace transform of $\cos(at)/t$

If someone could help me solve for $$\mathcal{L}\left\{\frac{\cos(at)}{t}\right\}$$ it would be great. Step-by-step I have so far: $$\begin{align}\int_0^\infty \frac{\cos(at)\space ...
0
votes
0answers
34 views

Complex inversion of a function

I am trying to find the function whose laplace transform is below using the complex inversion formula: $$ f(s)= \frac{se^s}{(s-2)^3}$$ My attempt below seems to be giving me the wrong answer but I'm ...
0
votes
0answers
10 views

Existence/Uniquness/Solution of a countably infinite system of linear ODEs.

Consider the following system of ODE's. \begin{align*} \frac{\mathrm{d}^2}{\mathrm{d}t^2} x_0 &= F(t) + x_1\\ \frac{\mathrm{d}^2}{\mathrm{d}t^2} x_i &= x_{i+1}-2x_i+x_{i-1}\, \quad ...
1
vote
1answer
38 views

Laplace transformations on a homogeneous ODE

$$y^{\prime\prime} - 3y^{\prime} + 2y = 0$$ $y(0) = 14$, $y^{\prime}(0)=0$, and using the Laplace transformation I'm trying to solve this IVP
0
votes
0answers
27 views

Matched pole zero discretization

There are several techniques to discretize continuous-time transfer functions to discrete-time transfer functions. Some of them, such as, zero-order-hold, forward euler or Tustin, are well known. ...
0
votes
1answer
46 views

Solving Second-order non-linear ODE, with fractional expansions

I am solving a differential equation related to fluid mechanics, a rigid air bubble rising towards the surface of a liquid. Doing all of the maths, I have come to this differential equation, which I ...
0
votes
0answers
16 views

Proving that if $f \in \mathcal{E}$, then $f' \in \mathcal{E}$ (same for $\mathcal{E}_q$)

In the context of ordinary differential equations, I'm trying to prove that if some function $f$ is an element of $\mathcal{E}$, which is the function space of all exponential polynomials, then $f'$ ...
0
votes
1answer
25 views

Laplace transformation on an exponential

Using the definition of Laplace transformation (and without using a table), how to find the Laplace transformation of $$ g(t)= \begin{cases} 0,&\text{if }0\leq t\leq 4;\\ e^{3t}&\text{if }4\le ...
2
votes
1answer
43 views

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 $N^{-s T}$ ...
-1
votes
2answers
27 views

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 ...
0
votes
2answers
50 views

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 ...