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

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Solving Differential equations with Laplace transform

$\displaystyle y''+4y'+3y=e^{-t}$, given $\displaystyle y(0)=y'(0)=1$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ $\displaystyle [s^2\bar y-sy(0)-y'(0)]+4[s\bar ...
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Solving simultaneous equations using Laplace transforms

$\displaystyle \frac{dx}{dt}+y=\sin t$ $\displaystyle \frac{dy}{dt}+x=\cos t$, given $\displaystyle x(0)=2, y(0)=0$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ ...
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Laplace transform - Heaviside algebra

I'm strugling with some algebra around a laplace transform with heaviside. The start function is $L(2tH(1-t)) + L(2H(t-1))$ so from this, I'm supposed to convert it to $L(2t) + L(2(1-t)H(t-1))$ ...
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Find the inverse laplace transform of $\displaystyle \frac{s}{a^2s^2+b^2}$.

Find the inverse laplace transform of $\displaystyle \frac{s}{a^2s^2+b^2}$ My Thoughts: Take $\displaystyle s^*=\frac{s}{a}$ and $b^*=\frac{b}{a}$ and divide numerator and denominator by $a^2$. ...
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How to find the Direct Discrete Laplace Transform of ${2n \choose n}$

Some time ago I developed a discrete version of the Laplace transform for the purpose of calculating sums and solve finite difference equations with constant coefficients. The notes below are a ...
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Heaviside Expansion Theorem with matrices

Is the Heaviside Expansion Theorem (HE) for the determination of inverse Laplace Transforms true for matrix expressions such that $\mathscr{L}^{-1}[\mathbf{P}(s)\mathbf{Q}^{-1}(s)] = \sum_i^n ...
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Determine tha Laplace transform using Heaviside fucntion

I want to determine the Laplace transform of the following function: $$f(t):t \mapsto \begin{cases}0, \quad t< 0 \\t, \quad 0\leq t \leq2\\2,\quad t>2\end{cases}$$ I have done it using standard ...
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Confused by a Laplace transform of $f(t)=t^ne^{at}$

Having looked at my lecture notes I was confused by the following part of a derivation of a Laplace transform for the function $\;f(t)=t^ne^{at} ,\quad n\ge0,\; a \in \mathbb{C}, \; f(t)=0 \;\forall ...
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12 views

Inverse Laplace transform, none factorable denominator

I am really stumpted on this problem and can't seem to figure out where to go from where I am. Can anyone give me some advice or hint where I should do next? Here is the problem: ...
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27 views

Using Laplace transforms to solve a convolution of two functions

Hi I have this problem where I need to take the convolution of functions and I am not sure if I got the right answer or something close so any advice or help would be very appreciated. So here is the ...
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20 views

Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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58 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
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35 views

Origin of Laplace Transform

Is the Laplace transform the continuous version of the infinite power series? $$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$ I learned this by watching this video lecture: ...
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31 views

Inverse laplace Transform of

How to calculate the inverse laplace transform of $\frac{\omega }{\left ( s^{2}+\omega ^{2} \right )( s^{2}+\omega ^{2} )} $ ?
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23 views

Inverse Laplace Transform partial fraction

While solving a second order differential equation, I have reached at a stage where I have to calculate the inverse laplace transform of $\frac{\omega ^{2}}{\left ( s^{2}+\omega ^{2} \right )( ...
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28 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
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22 views

Inverse Laplace Transform of a complicated function

Help with finding the inverse Laplace Transform of $$F(s) = \frac{1}{s\sqrt{M^{2}-s^{2}}}e^{\frac{\sqrt{N^{2}-s^{2}}}{s}}$$
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7 views

Maximum output value

I have the following transfer function: $\frac{sX_{2,r}(s)}{X_{2,r}^0} = \frac{s\omega_n^2R}{s^2 + 2\zeta \omega_n s + \omega_n^2}$ If I am correct, the signal $\dot{x}_{2,r}(t)$ is limited to the ...
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14 views

How did this function turn to the other through the laplace transform?

I was studying for my exam doing exercises, and in one of the questions where you had to use the laplace transform, they transformed this: $$X'(t)=AX(t)$$ into this:$$sX(s)-X(t=0)=AX(s)$$ where ...
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47 views

What is the $s$ in the Laplace transform?

I know the formula, which is $$F(s) = \int_0^\infty f(t) e^{-st}\,dt$$ but I don't understand what the $s$ is, and I've been searching everywhere and can't seem to find an answer, or maybe I'm just ...
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Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
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22 views

Laplace transform and Power series

As it can be read here, Discrete to Continuous Representations of Functions via Laplace Transforms? the Laplace transform is a continuous analog of a power series in which the discrete parameter n is ...
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50 views

Inverse Laplace Transformation of an exponential function

How one could find the inverse Laplace transformation of $\exp(-(b/(b+s))^k)$? Where both $b$ and $k$ are positive.
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45 views

Prove that some function is the solution of some equation

Show that $$x(t)=\sum_{n=0}^{\infty}\frac{(-1)^n(t/2)^{2n}}{(n!)^2}$$ is the solution of $$x*x=\int_{0}^t x(u)x(t-u)du=\sin t$$ My approach: I suppose that I have to use the Laplace transform. I ...
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Laplace Transform of $\frac{a}{2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t))$

I'm trying to prove that the Laplace Transform of $$\frac{a} {2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t)) $$ is $$\exp(-a\sqrt{s}); $$ from the definition of Laplace Transform we should compute ...
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Laplace Transform and differential equation

Question is d^2y/dt^2+3dy/dt+2y=u(t-1), where y(0)=0 and y'(0)=1 This is my working, is it correct? (For my 1st part of the answer)
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28 views

Laplace transform non exponential order.

There's an existance/uniqueness theorem which states: if function f is of exponential order, then the Laplace transform of f exsists and is unique. Since there's no theorem in my book which states ...
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Laplace transformation $y''+2y'+2y=3\sin x+\cos x$

Given$$y''+2y'+2y=3\sin x+\cos x$$ Transform to image region $$Y(s)(s^2+2s+2)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ $$Y(s)((s^2+2s+1)+1)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ ...
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47 views

Inverse Mellin transform of simple function

How should the following inverse Mellin transform integral be evaluated: $ f(x) = \displaystyle \frac{\alpha}{2\pi i}\int_{c-i\infty}^{c+i\infty}(\alpha ...
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Transfer Function non-zero initial conditions Laplace Transform

okay I know how to get transfer functions using the Laplace Transform assuming zero initial conditions but I would like to know how to deal with non-zero initial conditions. One of my maths books ...
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Explanation on how is simplified expression $\frac{s^2+3s+3}{2s^2+7s+7}$

This is done in the solution of exercise in order to make it possible to do inverse Laplace transform. Though I am not sure how is that done, so here it is: ...
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Confusion regarding bilinear transform.

I was reading my book where the z-transform of a signal is derived to be ${1-e^{-2bT}z^{-1}}$ . Then it goes on to say that by applying the bilinear transform we can get ...
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What is Wolfram Alpha returning when I ask it to calculate $\mathcal{L}[e^{x^2}](s)$?

As my instructor mentioned more times than could ever seem necessary, arguments of the Laplace transform have to have growth of exponential or less. I typed this "nonexample" into W|A and noticed ...
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Laplace Transform with sin and cos

Hi I am having trouble figuring out the solution of this Laplace transform: $$L_t{(u(t- \pi)(2\cos(t)-3\sin(3t))}$$ Where I am stuck if I am even on the right track is: $$L_t{(u(t- ...
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59 views

Laplace Transform, Unit Step Function

Hi I have been trying to do this Laplace Transform and cant seem to figure it out and was wondering if someone could point me in the right direction; here it is: $$L_t{(u(t-2)(2t^2-6t+5)})$$ What I ...
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Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
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41 views

Laplace transform of a differential equation

Given the Laplace transform \begin{align} \mathcal{L}\{g(r)\} = f(t) = \int_{0}^{\infty} e^{-tr} g(r) \ dr \end{align} can it be shown that the transform of the differential equation \begin{align} ...
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Solve Laplace Integral (3 factors) [closed]

Please provide steps to solve this integral: $$3\int^t_0{\sin{u}(t-u)e^{-(t-u)}du}.$$
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Solving an ODE using Laplace Transforms

$$y′′′′ + 2y′′ + y = \sin x$$ $$y(0) = y′(0) = y′′(0) = y′′′(0)= 0$$ After solving I got $y(s)=\dfrac1{(s^2 + 1)^3}$ for which I am unable to find the inverse Laplace transform. Please let me know ...
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31 views

Find Inverse Laplace Transform

I'm trying to calculate $$L_s^{-1}\left({\cfrac{2s+12}{s^2+9}}\right)=2L_s^{-1}\left({\cfrac{s+6}{s^2+9}}\right)$$ But I do not know how to go from here. I have noticed that the bottom does look ...
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39 views

How to find the inverse Laplace transform of this function?

$F(s) = \dfrac{s}{s+3}$ I can't find this term in any Laplace lists around
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Heat Equation derivative in terms of Laplace

If the heat equation is $ \frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$ Is the second derivative of u w.r.t t is the laplacian of the lapacian?
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Taking Laplace-Stieltjes transform to find virtual idle time in G/M/1 queue

I am reviewing some queueing problems from Gross and Harris, and had a question on problem 5.40 part b. The problem is stated as follows: Part B: Show that the stationary output of a $ G/M/1 $ queue ...
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analytical solution for linear 1st order PDE using laplace and seperation of variables

I am looking for the solution of the following pde: $\frac{\partial y(x,t)}{\partial t} = a* \frac{\partial y(z,t)}{\partial x} + b* y(x,t) + c$ and need help with the boundary and initial ...
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Inverse Laplace with $\ln$

How can I compute the inverse Laplace of 1) $\ln\left(\dfrac{s+1}{s-1}\right)$ 2) $\ln\left(\dfrac{s-1}{s}\right)$. Can someone please hep me to do these two problems
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29 views

Inverse Laplace transform of a rational function

Could you please help me the following Inverse Laplace problem $$\frac{2s^2+4s+3}{s(s^2+s+0.5)}$$ $F(t)$ is required.
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56 views

Laplace transform of $L({1-e^{-t}\over t})$

I have to find the Laplace transform of $$\mathcal{L}\left[\dfrac{1-e^{-t}}t\right],$$ then this is equivalent to $$\mathcal{L}\left[\dfrac{1}t\right]-\mathcal{L}\left[\dfrac{e^{-t}}t\right]$$ But ...
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Laplace transform of $f(t)/t$

For a function $f(t)$ Laplace transform is defined as $F(s)=\int_0^{\infty} f(t)e^{-st}dt$. I have to show the property that the Laplace transform of $f(t)\over t$ is $\int _s^\infty F(s')ds'$. ...
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How to compute transfer function from Laplace Transform

My system of interest has the following EOM (V is my input variable): $\ddot{x} = g - k_{1}V(t) + \dot{x}k_2$ Taking the Laplace with initial conditions of zero, I get: $s^2X(s) = \frac{g}{s} - ...
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30 views

Inverting Laplace transform

I am trying to solve the integral-differential equation: $$x'(t) + \int_0 ^{t} (t-s)x(s) ds = t + \frac{1}{2}t^2 + \frac{1}{24}t^4$$ With $x(0) = 1$ Taking the Laplace transform of this and using the ...