Questions tagged [laplace-transform]
The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.
0
votes
1answer
22 views
Variance of a Laplace Transform
I have a function $F(s)$ which is the Laplace transform of $f(t)$ (which is in itself a normally distributed random process), but I don't know what $f(t)$ is (this comes from solving a differential ...
0
votes
0answers
35 views
Finding $y(t)$ in a causal system given an input-output relationship
I want to find $y(t)$ in a causal system with input-output relationship
$$\frac{dy(t)}{dt} + 3y(t) = x(t)$$
where
$$x(t) = e^{2t} \cdot u(-t).$$
Here, $u$ is the Heaviside function.
To try and ...
0
votes
0answers
29 views
Taking the Laplace transform of the derivative of some function
I need to find the Laplace transform of $\dfrac{dh(t)}{dt}$ where $h(t) = e^{-100t} \cdot u(t)$ and $u(t)$ is the Heaviside function.
I have two ideas for solving this problem but I am unsure which ...
0
votes
4answers
37 views
Solving O.D.E and Initial Values Problem using Laplace Transform
I have this ODE:
$$
y'' + y =
\begin{cases}
\cos t, &\text{ if }0\le t \lt \pi\\
t-\pi,&\text{ if }\pi \le t \lt \infty
\end{cases}
$$
The initial values are:
$$
y(0)=0 \\
y'(0)=0
$$
I ...
1
vote
0answers
62 views
Inv. Laplace $\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$
What would be the inverse Laplace of the following function?
$\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$ ...
2
votes
1answer
32 views
Inverse Laplace transform of $F(s)=\frac{1}{s\sqrt{s+b\sqrt{as}\tanh{(\sqrt{as})}}}$ using complex integration
I want to find the inverse Laplace transform of
$$F(s)=\frac{1}{s\sqrt{s+b\sqrt{as}\tanh{(\sqrt{as})}}}$$
I tried to use the Bromwich integral
$$f(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\...
0
votes
1answer
19 views
Laplace Transform with Multiplied Terms
I have to take the laplace transform of $te^{3t}\cos(3t)$
So I started out with the property $L\{t^nf(t)\} = (-1)^n\frac{d}{ds} L\{f(t)\}$
Which yielded $-s^2-b^2+2s^2/(s^2+b^2)^2$. If I did it ...
0
votes
2answers
25 views
Solve system differential equation using Laplace transform
Question :
Solve system differential equation using Laplace transform :
$\cases{x'-3x+2y= \sin t\\4x-y'-y=\cos t\\x(0)=0,y(0)=0}$
My try :
Take Laplace we find :
$\cases{(s-3)X(s)=\frac{1}{s^...
0
votes
1answer
17 views
Numerical inverse Laplace
I have this function
\begin{equation}
F(s) = \exp\left(a s + \frac{b}{s+\gamma}\right),
\end{equation}
for which I need to compute the inverse Laplace transform, at least numerically. I can compute ...
1
vote
1answer
57 views
Get the equation of the plot of the response of the transfer function
In the case of permanent magnet DC motor position control, I wanted to rotate my motor shaft along a reference trajectory. Then I designed a piecewise function for reference and simulated the response ...
0
votes
0answers
20 views
Partial integro-differential equation using Laplace transform
Is it possible to solve the linear PDE analytically
\begin{equation}
\frac{\partial u}{\partial z} + a \frac{\partial u}{\partial t} + \int_{0}^{t} e^{-\beta (t-t')} u(z,t') dt'=f(z,t),
\end{equation}
...
0
votes
6answers
80 views
Find inverse Laplace Transform of : $\frac{1}{(s^2+a^2)^2}$
Question :
Find inverse Laplace of :
$$\dfrac{1}{(s^2+a^2)^2}$$
My try :
$$\dfrac{1}{(s^2+a^2)^2}=-\frac{1}{2s}\frac{\mathrm d}{\mathrm ds}\left( \frac{1}{s^2+a^2}\right)$$
I need use this ...
3
votes
1answer
75 views
Find inverse Laplace transform of : $\ln(\frac{s^2+a^2}{s^2+b^2})$
Question :
Find inverse Laplace transform of :
$$\ln \left(\frac{s^2+a^2}{s^2+b^2}\right)$$
My try :
I'm trying use this identity :
$f(t)=-\frac{\mathcal{L}^{-1}(\frac{dF(s)}{ds})}{t}$
...
0
votes
0answers
17 views
Solving integral equation via Laplace transform
I want to solve the following integral equation
$$\int_0^\tau \ddot{\Psi}(t-t')g(t')dt'= g(\tau^-)\dot{\Psi}(\tau-t)+g(0^+)\dot{\Psi}(t),$$
where $\Psi$ is a even function, $g(\tau^-)\neq g(0^+)$ ...
0
votes
1answer
39 views
Find Laplace transform of $t-\pi$
I am dealing with an Initial Value Problem of a step function:
$$
y'' + y =
\begin{cases}
\cos t, &\text{ if }0\le t \lt \pi\\
t-\pi,&\text{ if }\pi \le t \lt \infty
\end{cases}
$$
I am ...
0
votes
2answers
41 views
Inverse Laplace transform of $f(s)={\frac{1}{s^{3/2}}}$ using complex integration
I want to find the inverse Laplace transform of
$$f(s)={\frac{1}{s^{3/2}}}$$
Refer to the Laplace transform table, and I found that the result is
$$F(t)=2\sqrt{\frac{t}{\pi}}$$
But I do not know how ...
2
votes
2answers
44 views
Partial fraction expansion of $\frac{1}{(s+1)^{2}(s-1)(s+5)}$
I'm seeking a partial-fraction expansion of $A=\frac{1}{(s+1)^{2}(s-1)(s+5)}.$
I was solve equation differential using Laplace transform, but I need use partial fraction of $A$.
1
vote
0answers
29 views
Inverting a Laplace Transform with the Residue Theorem
I'm trying to invert the following Laplace Transform using the Residue Theorem:
$$
F(s) = \frac{K_1(\sqrt s)}{\sqrt s K_0(\sqrt s)}
$$
where $K_0()$ and $K_1()$ are the Modified Bessel Functions of ...
1
vote
2answers
38 views
Clasify the output $y(t)$ knowing that $x(t)=e^{-t}$, $t>0$
Given the transfer function $$G(s)=10\frac{s+1}{s^2-2s+5}$$ clasify the output $y(t)$ given the input $x(t)=e^{-t}$, $t>0$.
I know that $$y(t)=\mathscr{L}^{-1}[Y(s)]=\mathscr{L}^{-1}[X(s)G(s)].$$ ...
2
votes
2answers
55 views
Inverse Laplace transform of $\frac{π\cosh(\sqrt s)}{2 s^{3/2} \sinh(\sqrt s)}$ using complex integration
I want to find the inverse Laplace transform of
$$F(s)=\frac{π\cosh(\sqrt s)}{2 s^{3/2} \sinh(\sqrt s)}$$
using the Bromwich integral
$$f(t)= \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{π\...
2
votes
1answer
141 views
Asymptotic behaviour of Laplace transform
If the functions $x(t)$ and its derivatives $x'(t), x''(t), \ldots, x^n(t)$ are continuous* and $x(0^+) = x'(0^+) = x''(0^+) \ldots = x^{n-1}(0^+)=0$ ($0^+$ denotes the right side limit when the ...
3
votes
0answers
35 views
Solving the heat equation via Laplace Transform
Question:
Let $u=u(y,t)$. Solve the following PDE (heat equation) in the region $y,t>0$:
\begin{align}
& \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial y^2} = \cos(t) \\
& u(...
5
votes
2answers
81 views
What is the solution to linear ODE $\dot x = Ax + b$?
Suppose I have a system of linear ODE
$$\dot x = Ax + b$$ where $A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n$
I know that when $b = 0$, using Laplace transform we have $x(s) = (sI-A)^{-1} x(0)$...
1
vote
1answer
41 views
A proof for the Final Value theorem using Dominated convergence theorem
I'm going over the proof for the Final Value theorem using the Dominated Convergence theorem on Wikipedia, I don't understand how from the equation
$$sF\left(s\right)=\int_{0}^{\infty}f\left(\frac{t}{...
1
vote
0answers
31 views
Mittag-Leffler function and Fox-Wright function
I find the following identity in many special functions books without proof. This identity is called the Laplace transform of the Mittag-Leffler function with three parameters. The result is in the ...
1
vote
1answer
24 views
Laplace of the square root of trig functions?
I want to find the laplace of 1/rootcos(t)
Laplace calculators don't give an answer with this as an input. I know nothing about laplace, so can someone explain why this happens? Is it impossible to ...
6
votes
1answer
70 views
Alternative version of the Final Value theorem for Laplace Transform
Let $f:[0,\infty) \to \mathbb{C}$ be a continuous and bounded function such that the limit
$$\lim_{T\to\infty} \frac{1}{T} \int_{0}^{T}f(t)dt = d \quad\text{exists}.$$
Let $\hat{f}$ be the Laplace ...
1
vote
1answer
28 views
differentiation of Laplace transform solution
I am wondering if there is a solution to the differential equation (of sorts):
$$\frac{d}{ds}\mathcal{L}\left[y(t)\right]-\mathcal{L}\left[\frac{d}{dt}y(t)\right]=0$$
Using the fact that:
$$\frac{d}{...
1
vote
0answers
30 views
Laplace transform of a sum of Heaviside functions
The questions is, find:
$$\mathcal{L}\left[\sum_{n=0}^\infty(-1)^n\text{H}(t-n)\right]$$
I started by saying:
$$\mathcal{L}\left[\sum_{n=0}^\infty(-1)^n\text{H}(t-n)\right]=\int_0^\infty\sum_{n=0}^\...
1
vote
2answers
39 views
Self convolution function?
Given two positive real functions $h$ and $g$ and,
$$g(t) \simeq \int\limits_0^t h(\tau) \cdot g(t-\tau) d\tau$$
i.e., $g$ asymptotically approximately equals the convolution of itself and another ...
1
vote
1answer
35 views
Finding the Inverse Laplace Transform of $-\frac{1}{s(s+1)}+\frac{(s+1)(s+2)}{s\left((s+1)^2+1\right)}$
I am trying to find the inverse Laplace transform of $$F(s)=-\frac{1}{s(s+1)}+\frac{(s+1)(s+2)}{s\left((s+1)^2+1\right)}.$$
I proceeded as follows:
\begin{align}
F(s)&=-\frac{1}{s(s+1)}+\frac{(s+...
0
votes
2answers
78 views
Inverse Laplace transform help- inverse heat conduction
I've been stuck on an inverse Laplace transform for my research. Would be greatly appreciative of any help solving the inverse Laplace transform of
$$
\overline{T}=\frac{2}{k_{1}} \frac{1}{s} \left(\...
0
votes
1answer
43 views
Using Laplace transform to find $\int_{0}^{\infty}\frac{\sin^2 t}{t^2}dt$ [closed]
Find
$$\int_{0}^{\infty}\frac{\sin^2t}{t^2}dt$$ using Laplace transforms.
0
votes
0answers
34 views
Unilateral Laplace Transform Uniqueness and Dirac Delta “function”
Answering this question, I've tried to alert the OP about the misleading definition of $\delta(t)$ - used in one of the answers - as:
$$
\delta(x) = \left\{\begin{array}{cc}
\infty & x = 0 \\
0 &...
1
vote
3answers
101 views
What is the correct answer for $\mathcal{L}^{-1}\left(\frac{p^2}{(p-3)^2}\right)?$
What is the correct answer for $$\mathcal{L}^{-1}\left(\frac{p^2}{(p-3)^2}\right)?$$
Using partial fraction technique I got the answer as: $\delta(t)+e^{3t}9t+6e^{3t}$ and uing shifting thorem for ...
1
vote
2answers
51 views
Laplace Transforms with Non-repeated Irreducible quadratic factors
I'm solving the following differential equation with a Laplace Transform:
$$x″+ 9x = \cos(t) + \delta(t-\pi)$$
The initial conditions are that x(0) and x'(0) are equal to 0.
After after the ...
0
votes
1answer
28 views
what is the Laplace transform of exponential function of dependent variable?
Given the following equation,
$\dot x(t) = e^{x(t)}$
The Laplace transform of $\dot x $ is $ sX(s) - x(0) $.
a) What is the Laplace transform (natural transform as well) of $e^{x(t)}$ ?
b) or ...
0
votes
0answers
24 views
Diffusion equation solution using Laplace transform
Consider the operator $L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t} $ with domain : $D(L)={{u} \in \Re \times [0,+\infty ):u(x,0)=g(x), \forall x\in \Re , \underset{x\...
0
votes
3answers
55 views
Show that the inverse Laplace transform of F(s) = logs is given by f(x) = -1/x
Here's what I did, but I'm not sure if it's right/ valid.
F(s) = log(s)
L(xf(x)) = -1
So, L(xf(x)) = (-1)(1/s)
Therefore, xf(x) = -1
So, f(x) = -1/x
0
votes
1answer
24 views
Laplace inverse as a double integral
Show that $$\mathcal{L}^{-1}\left[\frac{f(s)}{s^2}\right]=\int_{0}^{t}\int_{0}^{x}F(x)dxdy.$$
I tried using the formula $$\mathcal{L}^{-1}\left[\frac{f(s)}{s}\right]=\int_{0}^{t}F(x)dx.$$
2
votes
1answer
43 views
How to calculate the Laplace transform of $\sum_{k=0}^{+\infty}(-1)^{k}\delta (t-k)$?
On my midterm, I had the following question:
Calculate the Laplace transform of $$\sum_{k=0}^{+\infty}(-1)^{k}\delta (t-k)$$
I was wondering how I should calculate it. I know that the transform ...
1
vote
1answer
33 views
Do Inverse Laplace transforms satisfy the Convolution Theorem too?
The Laplace transform of a function $f$ is defined as
$$
F(s) \ = \ \mathcal{L}[f](s) \ = \ \int_0^\infty dt \ f(t) e^{-st}
$$
Whereas we can write the inverse Laplace transform in terms of the ...
3
votes
1answer
143 views
Breaking up a Convolution Integral
For a diffusion problem in a semi-infinite domain with a transient boundary condition, the temperature profile can be obtained from Duhamel's Principle as
$$
\theta(r,t)= \int_0^t \frac{\partial \phi}...
0
votes
0answers
19 views
Does a time function has several Laplace transforms
I was wondering: Does a given time function has several Laplace transforms?
For example, take a simple exponential function:
$f(t)=e^{-at}$
When the inverse Laplace transform was done with the ...
0
votes
0answers
12 views
Decompose Inverse Cubic into Partial Fractions for Inverse Laplace Transform
I'm trying to perform an inverse Laplace transform on the transfer function:
$H(s) = \dfrac{15}{s^3+6s^2+15s+15}$
I know the roots of this equation and for simplicity's sake, I've let the factors ...
1
vote
1answer
30 views
Integral of Dirac-delta function from convolution theorem
In a question I have been lead to use the convolution theorem to find the inverse Laplace transform, as shown below:
$$\omega(t)=\mathscr{L}^{-1}\left[e^{-bs}\frac{s}{s^2+a^2}\right]$$
From the ...
0
votes
1answer
41 views
Output response from closed loop transfer function using MATLAB
This transfer function is to control the position of Permanent Magnet DC motor. I was able to get the transfer function and now I need to analyze the output for the tuned closed loop for a given input ...
-1
votes
0answers
42 views
Inverse Laplace transform of $F(s) = \exp(-a\sqrt{s})/s$, $a > 0$
Show that the inverse Laplace transform of F(s) = $e^{-as^{1/2}}/s$, $a > 0$, is given by
$$f(x) = 1 - \frac{1}{\pi}\int^{\infty}_{0} \frac{\sin(a\sqrt{r})}{r}e^{-rx}dr$$
Note that the integral ...
1
vote
1answer
30 views
What kind of integral is the one in Laplace transform?
In a book about circuits I found a chapter where it is defined the Laplace transform of a function $f:\mathbb{R}\rightarrow\mathbb{R}$ as $$L[f(t)]=\int_0^{+\infty}f(t)e^{-st}dt$$ where $s\in\mathbb{C}...
1
vote
0answers
37 views
How to find the laplace transform of $\cos(\sqrt t)$?
I tried solving for the transform using the same method the book uses to find laplace transform for $\sin(\sqrt t)$ which is, by writing the Maclaurin's expansion for $\sin(\sqrt t)$ and then using ...
