1
vote
0answers
23 views

'Deriving' the Laplace Transform from the z Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
0
votes
1answer
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} ...
0
votes
2answers
58 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 ...
2
votes
2answers
186 views

Prove that the Laplace trasform is a Linear trasformation

Could you help me prove that the Laplace Trasform is a Linear trasformation? Thank you.
1
vote
0answers
38 views

Laplace transform of $\sin(x)$

I am confused with Laplace transform of $\sin(\theta)$. For example, what is the LT of $A \sin(x(t))=Bx''(t)$ ($x$ is second order), $A,B$ are constants.
0
votes
0answers
37 views

Laplace Transform of $\sin(t-3)$

I wanted to compute the Laplace Transform of $\sin(x-3)$ using the shift rule: $\mathcal{L}(f(t-a)) = e^{-as}\mathcal{L}\left(f(t)\right) \Rightarrow \mathcal{L}(\sin(t-3)) = e^{-3s}\mathcal{L}(\sin ...
1
vote
1answer
83 views

Evaluate $\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$

Prove or disprove that$$\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$$ for any $a>1$. I came across with this integral evaluating inverse ...
0
votes
1answer
41 views

Find the poles and residues in an awkward Laplace inversion

Assume that part c) has been proved and ignore parts c) & d). To invert the Laplace transform we would do $\displaystyle u(x,t)=\frac{1}{2\pi ...
2
votes
2answers
59 views

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} = ...
0
votes
1answer
55 views

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 ...
7
votes
1answer
58 views

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 ...
2
votes
1answer
74 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 $2n\pi<t<2(n+1)\pi$ ...
1
vote
2answers
50 views

Laplace transform of $\sin^2(\omega t)$

What is the Laplace transform of the function $\sin^2(\omega t)$
2
votes
2answers
67 views

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) ...
1
vote
0answers
25 views

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 ...
2
votes
2answers
58 views

Laplace transform of a piecewise function

I'd like to compute the Laplace transform of the following function: $$f(t) = \begin{cases} 0,& \mbox{if} \quad 0 \leq t \lt \pi \\ \sin(t), &\mbox{if} \quad t \geq \pi \end{cases}$$ Could ...
0
votes
0answers
14 views

A question about Parseval's formula.

In operational calculus there is Parseval's theorem, which states that if $ f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p) $ and both $ F(p) $ and $ \Phi(p) $ are analytical in $ Re p \geq 0 $, ...
0
votes
2answers
29 views

applying two Laplace properties on same function

For example, the Laplace transform of $(t - 3)\cdot u(t-3)$ I'm confused about how to apply the two Laplace properties (multiplication of t and time shift). Do I apply one property first then the ...
4
votes
1answer
181 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
0
votes
1answer
67 views

Representation of heaviside step functions

Can the heaviside step function, $u(t)$ be represented like so: $$u(t)=\frac{1}{2}\left(\frac{|x|}{x}+1\right)$$
4
votes
1answer
82 views

Inverse Laplace

Hi how to verify the following I tried substitution and integration by parts but can bot figure it out.. $$\int_0^{\infty} \exp(- \lambda t ) \frac{x}{\sqrt{2\pi t^3}}\exp(-\frac{x^2}{2t}) dt = ...
1
vote
0answers
109 views

Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) ...
1
vote
1answer
50 views

Is that possible to solve the following differential equation by using laplace transform?

Is that possible to solve the following differential equation by using laplace transform? $$y''-t^2y'+y=e^{2t};\quad y(0)=1,\quad y'(0)=1$$ ?? I knew that if there is coefficient except $1$ with ...
0
votes
2answers
150 views

Find the inverse Laplace transformation of $\frac{e^{-s}}{s+2}$

My question is: Find the function $f(t)$ that has the following Laplace transform $$F(s)=\dfrac{e^{-s}}{s+2}$$ Thanks . my try:I have find this Find the inverse Laplace transformation of ...
0
votes
1answer
61 views

Laplace Transformation spring question

Here is the question: http://i.imgur.com/XAH2UnX.jpg I can't seem to get the answer. Are those values in the writing like 1N/m even relevant? Can someone give me some direction? Thanks!
0
votes
1answer
724 views

Laplace Transform Involving Heaviside Step Function

I'm trying to find the Laplace transform of $7 e^{-3t} u(t-3)$, where $u$ is the heaviside step function. However, we've never really gone through what the Laplace transform of the heaviside step ...
1
vote
1answer
70 views

Laplace Transform using t-shift

$$f(t)=\begin{cases}cos(πt), & 1\leq t < 4 \\ 0, &elsewhere \end{cases}$$ Okay, I attempted to write it in terms of step functions and I got $$ f(t) = cos(πt)u(t-1)-cos(πt)u(t-4)$$ But ...
4
votes
1answer
201 views

Inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$ (Bromwich integral)

I am looking for the inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$. I started by using the general formula of the Bromwich integral: $\displaystyle \lim_{R\to\infty} \int_{a-iR}^{a+iR} ...
0
votes
1answer
83 views

Laplace Transform using t-shift (second shift)

$$f(t) = tu(t-π)$$ I know I have to get t in terms of $$(t-π)$$ and to do that I have done $$ t = a(t-π) + b$$ $$ t = at-aπ + b$$ $$ t = (a-π)t + b$$ $$ (a-π) = 1$$ and $$b = 0$$ Then I think I ...
1
vote
1answer
58 views

Computing the inverse Laplace transform of this?

What's the correct way to go about computing the Inverse Laplace transform of this? $$\frac{-2s + 1}{(s^2+2s+5)}$$ I Completed the square on the bottom but what do you do now? $$\frac{-2s + ...
2
votes
3answers
189 views

Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
6
votes
1answer
344 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
0
votes
2answers
49 views

More Laplace! - help needed

Here is the exam question that I am practicing: I have completed the first two parts to this question (thankfully to stackexchange) Laplace question - help needed Laplace question continued ...
0
votes
2answers
87 views

Laplace question continued (partial fractions)

Last night I attempted and successfully finished (with the help of stackexchange) the first part to this question on laplace transformations: Laplace question - help needed The second part to this ...
1
vote
2answers
66 views

Laplace question - help needed

I am currently studying the Laplace transformation and came across this question: I have no idea of how to start this and am completely lost. If anyone could help I would be really grateful. ...
1
vote
2answers
110 views

Laplace Transformation Applications

In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations. What kind of equations do Laplace ...
4
votes
2answers
406 views

Find the inverse Laplace transformation of $\dfrac{s+1}{(s^2 + 1)(s^2 +4s+13)}$

My question is : find the function $f(t)$ that has the following Laplace transform $$ F(s) = \frac{s+1}{(s^2 + 1)(s^2 +4s+13)} $$ thanks
1
vote
2answers
64 views

Laplace transform of $f(t)=10te ^{-5t}$

Find the Laplace transform of $$f(t)=10te ^{-5t}$$
3
votes
3answers
127 views

Calculate the next Inverse laplace transform

This question may be very basic, but I dont know how to get the next inverse laplace's transform: $${\scr L}^{-1}\left\{\frac{1}{(s-1)^2}\right\}$$ I can only use these two formulas: ${\scr ...
-2
votes
1answer
91 views

Prove that L[f' ' ](s)$ = $sL[f](s)

Can anyone prove this question ? Let $f$:$\mathbb{R}$$→$$\mathbb{C}$ be continuous function such that $f$$(0)$ $=$ $0$ and that $f'$ be a piecewise continuous function and absolutely integrable on ...
0
votes
1answer
250 views

Using Convolution Theorem to find the Laplace transform

In previous questions I have used Laplace transform to find the inverse Laplace transform. I have worked through this work booklet ...
2
votes
3answers
141 views

Find Laplace Transform of the function

$$f_T(t)=\begin{cases}2, & 0\leq t < T \\ 1, & t\geq T \end{cases}$$
1
vote
1answer
75 views

I need to find a Laplace Transform but have problems

Using Laplace transform find: $e^{-10t}u(t)$. According to the Laplace table, that is equal to $1/s+10$, but how can I prove it?
2
votes
1answer
111 views

Functional form of a series of a product of Bessels

This question arises from my answer to an inverse Laplace transform question. The result I got took the form $$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) ...
2
votes
0answers
88 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
2
votes
0answers
116 views

A rational integral with exponential denominator

Prove that: $$\int_{-\infty }^{+\infty }{\frac{{{x}^{4}}\text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
0
votes
1answer
52 views

A improper integral on expontential

Evaluate: $$\int_{0}^{\infty }{\frac{\left( 1-{{\text{e}}^{-px}} \right)\left( 1-{{\text{e}}^{-qx}} \right)\left( 1-{{\text{e}}^{-rx}} \right)}{{{\text{e}}^{x}}}}\text{d}x,\ \ \ p>0,\ q>0,\ ...
0
votes
0answers
76 views

About Laplace transform

I dont understand the following working, why the integral becomes double integral? $$\begin{align} & \ \ \ \int_{0}^{1}{{{\left( \frac{1}{\ln x}+\frac{1}{1-x} ...
1
vote
1answer
52 views

Convergence integral causal function

I have an exercise where there is the following given: $f$ is a causal function. $f$ is Laplace transformable:$\int_{0}^{\infty} f(t)e^{-zt} \, dt = L(z) $ with $Real(z)> -1$ I have to ...
3
votes
2answers
976 views

Laplace transform of $t \cos(t)$ by definition

I want to find the Laplace transform of $t \cos(t)$ by the definition $$\int e^{-st} t \cos(2t)dt$$ The solution manual just say try the $$u = t, dv = e^{-st} \cos(2t)$$ I use the integration by ...