The Laplace transform is a widely used integral transform, similar to the Fourier transform.

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Solving a differential equation using the laplace transform involving convolution

The problem is the following The thing that puzzles me here is the integral on the right hand side, so: How to take the laplace transform on the right hand side? Any help to get me going would be ...
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Solving an integral using Laplace transform and inverse Laplace transform

I want to solve this integral equation using Laplace: $$ Y(t) + 3{\int\limits_0^t Y(t)}\operatorname d\!t = 2cos(2t)$$ if $$ \mathcal{L}\{Y(t)\} = f(s)$$ then, $$ f(s) + 3 \frac{f(s)}{s} = ...
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Fourier-Laplace Transform of Heaviside Step function multiplied to Sine

In a Advanced Solid State lecture I encountered the following assertion- Fourier Transform of $\Theta(t)\sin(\omega_0 t)$ is ...
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Demonstration with Laplace

How can I demonstrate this? If $F(t)$ is a periodic function with a period of $T>0$, then $$ \mathcal{L}\{F(t)\} = \frac{\int\limits_0^T e^{-st} F(t)\operatorname d\!t}{1-e^{-sT}}\operatorname ...
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inverse laplace using partial fractions and completing square

what is the inverse Laplace transform of this equation $$\frac{1}{(s+1)(s^2+s+1)}$$ I know that completing the square for the quadratic term is required to avoid complex roots and then I need to use ...
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Integral equation solve using Laplace transform

How can I solve this integral equation using Laplace transform? $${\int\limits_0^{\infty}\ }\frac{e^{-t}(1-\cos t)}{t}\operatorname d\!t$$ Knowing that $$ \mathcal{L}\{\cos t\} = \frac{s}{s^2+1} $$ ...
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Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
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Inverse Laplace Transform of $ \left(\frac{1-s^{1/2}}{s^2}\right)^2$

I found this question in my N.P Bali's Engineering Mathematics 7th Edition. I could not find any solved questions related to this. How can I find the Inverse Laplace Transform of : ...
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Finding the inverse Laplace transform of $\frac{s^2-4s-4}{s^4+8s^2+16}$

$$F(s) = \frac{s^2-4s-4}{s^4+8s^2+16}$$ My work is as follows, $$\frac{s^2-4s-4}{(s^2+4)^2}=\frac{s^2+4}{(s^2+4)^2}-\frac{8}{(s^2+4)^2}-\frac{4s}{(s^2+4)^2}$$ The inverse laplace of the first term ...
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integro-differential equation with application in quantum mechanics

I am trying to solve for the time dynamics for a simple quantum system (two-site system with sinusoidal coupling and a decay parameter on one site) and the math is looking not so simple. Here is the ...
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Laplace transformation: second shifting theorem

I know the answer is $1/(s^2) +e^-6s (2/s^3 -14/s -1/s^2 )$, but can anyone tell me how to evaluate the solution? I really get stuck.
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laplace transform with heaviside step function

There are some similar questions around but they didn't help me much. I've got a graph that I'm supposed to perform a laplace transform on. From $0<t<2$, it's a ramp function from $f(0)=0 to ...
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77 views

Solving a differential equation using Laplace transform

The problem has two parts: 1. Solve the initial value problem: $$ y''+y=\sum_{j=0}^\infty \delta_{2j\pi}(t) $$ with the initial conditions: $y(0)=y'(0)=0$ 2.Show that if $2n\pi<t<2(n+1)\pi$ ...
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Why Laplace transfrom uses Exponential function

I probably don't have enough background to post this question, but I am very curious about it. The way I think about the Laplace transform now, the Laplace transform multiplies a given signal by ...
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64 views

Laplace Transform Piecewise Function

I've never seen these types of bounds on a piecewise function of a Laplace transform before, can someone help explain how to solve this problem, particularly the Laplace transform of g(t)? Thanks in ...
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What Laplace transformation is used for this?

I cannot see the transition here. I was unable to find what Laplace transformation was used here.
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laplace transform of a sine function

I'm a little confused about how to find Laplace transforms of a sine function when it is a function of time. As in, suppose the function is $x(t)=\sin(at)$ , then I can proceed to get ...
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How to draw a diagram for the following PDE

Subject: Partial Differential Equations. Here are the details of the question: $$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$ ...
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What does “Formulate the system of equations for a finite difference discretisation of the problem” mean?

Subject: Partial Differential Equations. Here are the details of the question: $$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$ ...
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Laplace transform of $\sin^2(\omega t)$

What is the Laplace transform of the function $\sin^2(\omega t)$
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Expressing function in terms of unit step function

I have trouble expressing functions in terms of the unit step function, if someone could explain how it works that would be great. For example - $g(t) = t^2$ when $0 \le t < 2$ $4$ when ...
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Why does the Final Value Theorem not hold for a transfer function with more than one pole at the origin?

The Wikipedia article on the Final Value Theorem states the following for cases where it does not hold: There are two checks performed in Control theory which confirm valid results for the Final ...
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Laplace transform with initial value problem $y''+4y=12\sin(2t)$.

Using Laplace transforms solve the initial value problem. $$y''+4y = 12\text{sin}(2t); \qquad\qquad y(\pi)=-3, \quad y'(\pi)=-3$$ I have begun with writing: $\mathcal{L} (y'') = s^2y(s) -s y(\pi) ...
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Laplace transform of piecewise continuous function

$$f(t) =\begin{cases}t^2 & 0 \le t < 3,\\ 9& t \ge 3\end{cases}$$ Show that $f$ is of exponential order. Express $f$ in terms of the unit step function. Find Laplace transform of ...
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Laplace transform of convolution with no function of t

Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming. Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$ My Problem: To treat this as ...
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Proof that laplace's equation is rotationally invariant using chain rule

Suppose $(x, y)$ and $(p, q)$ are coordinates in the plane related by rotation around a fixed point $(a, b)$, as follows: $$\begin{bmatrix} p\\ q\end{bmatrix} = \begin{bmatrix} \cos(t) & -\sin(t) ...
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$y''+2y'+5y=0$, initial value problem with Laplace transform?

here is the question: $$ {\rm y}''\left(t\right) + 2\,{\rm y}'\left(t\right) + 5\,{\rm y}\left(t\right) = 0; \qquad\qquad {\rm y}\left(0\right) = 2\,,\quad {\rm y}'\left(0\right) = -1. $$ ...
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26 views

Find Laplace Transform of the following function

How do I find the Laplace transform for the function: $f(t)=t, 0 \leq t \leq 1$ and $2-t, t \geq 1$ I tried looking up the process online, but it remains unclear to me. Thanks in advance!
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Laplace transform, when $s \rightarrow \infty$

I'm reviewing lecture notes on Laplace Transform and there's one step that I don't understand: Find the solution to: $$x y'' + y' + xy = 0, y(0) = 1, y'(0) \mbox{ finite}$$ Taking the Laplace ...
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Solving a non-linear integro-differential equation

I am trying to solve the following equation $$ f^2(x) - g^2(x) = \alpha\int_0^x f(u) (x-u)du $$ For $\alpha=0$ we get $f=g$. I would like to see how the solution moves away from $g$ when I increase ...
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s-plane and fourier transform, together in 3d space.

I dont understand how can varying the real part in the s-plane make the amplitude in the fourier plane go to infinity. Lets say the pole is at -3 + -j for example.. Then the laplace transform is the ...
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Inverse Laplace transform of $s^{k}$

How can I find the inverse Laplace transform of $s^{k}$ where $k$ is non-integer and negative? I know that $$\mathcal{L}^{-1}[s^k] = \frac{1}{2\pi i}\int e^{st} s^k ds$$ and since we have ...
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Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
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Laplace Transform of tsin(at) using only the definition

Hello I' am stuck on how to get the final result of the laplace transform of $f(t)=tsin(at)$using (a is a constant) only the definition of $$\int_0^{\infty}f(t)e^{-st}dt$$, I know $sin(at)= {1 \over ...
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Laplace transform of a majorated function

I have the following problem. I have an analytic function and I want to show that it is majorated by a convenient function. To do that, it is very helpful to solve the transformed equation. I have a ...
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Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
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Solve second order differential equation with Heaviside function using Laplace transform

The equation is: $$y'' + 3y = u_4(t)\cos(5(t-4)), \quad y(0) = 0, \quad y'(0) = -2$$ Here $u_4$ is the Heaviside function with activation switch at $t=4$. I can get all the way to the partial ...
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Using Laplace Transform to solve a equation with piecewise function

Using Laplace Transform to solve$$y''+4y=f$$ Where $y(0)=0, y'(0)=-1,$ and:$$f(t)=\begin{cases}\cos(2t)&\text{if $0\le t \lt \pi$}\\0 &\text{otherwise}\\\end{cases} $$ Do I need to solve the ...
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Take the Laplace Transform

Take the Laplace transform of $$ \int_{0}^{t}x^2(x-t)^4 \cos(x)dx .$$ I'm not quite sure where to start...
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Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
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Laplace transform using the definition

Find the Laplace of the given function using the definition $$f(t)=tsin(t)$$ I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there ...
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Laplace Transform of an integral

Find the Laplace transform of $$f(t)=t\int_0^{t} \tau e^{-\tau}$$ $L(f)(s)$= ?? My thought is that I can change the $\tau$ to $t$ by Transforming the integral to get $$t/s*L[t*e^{-t}]$$ But ...
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Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
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Inverse Laplace Transform of Reciprocal Quadratic Function

Starting with the the equation: $$I(s)=\frac{6}{Ls^2 + Rs + \frac{1}{C}}$$ I need to find what $i(t)$ is by doing the inverse Laplace transform. I need to do some algebra to put it in a form that ...
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Laplace transform of $f(t)$ multiplied by $t^n$

How to prove that it is $(-1)^n\frac{d^n}{ds^n}F(s)$
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complex integral of z to the power alpha

I would like to perform an inverse laplace and at some point of the calculation I have to compute this integral $$\int_{\gamma-i\infty}^{\gamma+i\infty} z^{(1+n)\alpha-1}e^{z} \frac{dz}{2\pi i}$$ ...
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Laplace transform.

This is a past exam question. I'm having a bit of trouble at finding the inverse laplace transform of the following function. Any help would be great. $$\frac{s^2+1}{(s^2+4s+5)^2}$$ Thanks for ...
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Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
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Find the solution of the IVP using Laplace transforms

The equation is as such: $y''+y=t\sin t$; $y(0)=1, y'(0)=2$ I took the Laplace transform of both sides to yield $F(S)(s^{2}+1)-(s+2)=\frac{-2s}{(s^{2}+1)^{2}}$, and then ...
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Laplace Transform of a rounded function (or an infinitely discontinuous function)

Working on an assignment today, I thought of a problem I haven't been able to solve and haven't been able to find any solutions for online. What would the Laplace Transform of a rounded function be? ...