Laplace Definition and 1000 Threads

With a Laplace transform, we can remember common set ups; for example, \[ \mathcal{L}\{e^{-at}\} = \frac{1}{s + a}. \] When it comes to the inverse Laplace transform, I can only find the tables to remember in a book. However, if we go back to the Laplace transform, we can always do \[...

Homework Statement Consider the Laplace’s equation, ∆u(r,θ) = 0, inside the quarter-circle of radius 2 (0 ≤ θ < π, 0 ≤ r ≤ 2), where the boundary θ is insulated, and u(r,\theta/2)=0 Show that the insulated boundary condition can mathematically be expressed as \frac{\partial u}{\partial...

The typical way to evaluate $ \displaystyle \int_{0}^{\infty} \frac{\cos mx}{a^{2}+x^{2}} \ dx$ is by contour integration. In a recent thread I evaluated that integral using the Laplace transform. http://mathhelpboards.com/analysis-50/advanced-integration-problem-9129.html#post42551My...

The amplifier and filter of FIGURE 4 can be represented by the equivalent circuit of FIGURE 6. (Figs are in image 1 and 2 attached) (i) Draw the Laplace equivalent circuit and hence derive the equation for vout(s) in terms of R, L and C for a step input. (ii) Attempt, using the component...

Hello, As of recently, I've been working with Laplace transforms and have a question about their relationship to solving differential equations. I know the definition of the laplace transform and I know that a function is essentially being transformed from the time domain to complex...

Hi Guys, I have an expression that i am struggling to manipulate into a laplace transform. This expression should fit one or a combination of the common transform pairs. I believe the transform the expression should be fitting is either a unit step 1/s a unit ramp 1/s^2 an exponential 1/s+a...

I have been trying to derive the Laplace in spherical co ordinates. I have attached a file which has basic equations. I am trying to get the following equation. d(phi)/dx= -sin(phi)/(r sin (theta)). I have also attached the materials I am referring to. Can someone please help me derive...

Hi, I would like to find the inverse Laplace transform for 11/(s^2+16)^2 I have tried to expand it using the following partial fraction decomp to find the constants and take the inverse Laplace but this did not work C1(s)+ C2/(s^2+16) + C3(s)+C4/(s^2+16)^2 Does anyone have any suggestions?

Homework Statement Find: Inverse Laplace for x(t)= (e)^-5t*(t)^4 using laplace table and laplace properties. Homework Equations The Attempt at a Solution Well, I have been working on this problem for a few days now and cannot seem to figure it out. The two functions are not...

Homework Statement Hi I would like to know how to expand 20/2s(s^2+9) in order to find the inverse Laplace Transform of the function K(s) to gt k(t). The (s^2+9) term in the denominator is throwing my calculations off for the constants because of the s^2 term. Homework Equations...

Hello, I have a relatively simple question. after being unable to find it through google I have decided to ask you guys if you know what the Laplace transform of a unit step function that looks like this would look like Us(t-2) From tables, the Laplace transform for a regular units step...

Homework Statement Is there a way to evaluate L^{-1}(\frac{F(s)}{s + a})? I'm sure if it can be evaluate. Homework Equations The Attempt at a Solution

Given the transfer function \[ H(s) = \frac{Y(s)}{U(s)} = \frac{1}{s + 1} \] and \[ u(t) = 1(t) + 1(t - 1). \] How do I find U(s)? I know I take the Laplace transform of u(t) but with the two step functions how can this be done? The Laplace transform of the step function is \(\frac{1}{s}\)...

I can't seem to part of an inverse Laplace transform correct. \begin{align*} f(t) &= \frac{6}{5}\mathcal{L}^{-1}\bigg\{\frac{1}{s + 2}\bigg\} + \frac{3}{5}\mathcal{L}^{-1}\bigg\{\frac{3s - 1}...

Homework Statement Find the Laplace transform of f(t) = t \forall 0≤t≤T, 0 otherwise Homework Equations The Attempt at a Solution I write the function as tu(t)-t*u(t-T) That is turn on the function t at t=0 and turn the function t off at t=T. It seems to be right to me...

Hi~ I recently solve a Laplace transform problem as following L[int{t,0}cosh(t'-1)U(t'-1)dt']=? U(t'-1) is the unit step function(=1 for t'>1, =zero otherwise) According the standard Laplace formula : (1)L[cosh(t-1)U(t-1)]=exp(-s)*s/(s^2-1); (2)L(int{t,0}f(t')dt')=F(s)/s...

Homework Statement Part(a): State condition for laplace's to work. Find potential in space between electrodes. Part (b): Find potential inside cylinder and outside. Part (c): How would the answer change is one pair is kept at potential 0 while other V0? Homework Equations The Attempt at a...

I have a probability distribution as follows: \begin{equation}p_j(t)=\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{1,k}e^{ij\tilde{k}}\right|^2+\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{2,k}(t)e^{ij\tilde{k}}\right|^2\end{equation} where...

Prove the following Suppose that $f$ is piecewise continuous on $$[0,\infty) $$ and of exponential order $c$ then $$\int^\infty_0 e^{-st} f(t)\, dt $$ is analytic in the right half-plane for $$\mathrm{Re}(s)>c$$

I posted this in the homework section, but I haven't received any help, so hopefully putting it in this section won't be an issue. I'm trying to compute the integral $$ \int_0^ \infty \frac{ \cos xt}{1 + t^2} \, dt $$ using the Laplace transform. The first thing that catches my eye is the 1...

Homework Statement $$ \int_0^\infty \frac{\sin xt}{x} \, dt $$ Homework Equations The Attempt at a Solution $$ = \int_0^\infty L(\sin xt) \, dp $$ $$ = \int_0^\infty \frac{x}{p^2 + x^2} \, dp $$ $$ = x \int_0^\infty \frac{dx}{p^2 + x^2} \, dp $$ p = x tan theta: $$ = x \int_0^{\pi/2}...

Homework Statement Find $$ L^{-1} \left[ \frac{1}{ (p^2 + a^2)^2} \right] $$ Homework Equations $$ L [ x \cos ax ] = \frac{p^2 - a^2} { (p^2 + a^2) ^2 } $$ The Attempt at a Solution I have no idea. Any thoughts?

Homework Statement Why I am getting wrong answer related to this Laplace Transforms problem? According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get (e^{-st}/(s^{2} + a^{2}))(a sin at - s cos at)Homework Equations ∫e^{-st}cos at dt The...

Here is the DE: Δu(r,θ)=0, 1 ≤ r ≤ 2, 0 ≤ θ ≤ pi and here are the Boundary Conditions: u(1,θ)=sin(θ), u(2,θ)=0, u(r,0)=0, u(r,pi)=0 Based on the Boundary Conditions I believe this is half of an annulus. Using the 2D Laplace equation for polar coordinates, find the solution u(r,θ). I've...

Homework Statement Use Laplace transforms to solve the initial value problems. ##y''+4y=0;## ##y(0)=5;## ##y'(0)=0## Homework Equations The Attempt at a Solution $$y''+4y=0$$ $$L(y'')+L(4y)=L(0)\implies s^{2}Y(s)-sy(0)-y'(0)+4(sY(s)-y(0))=0\implies s^{2}Y(s)+4sY(s)-5s-20=0$$...

Homework Statement y''+6y=f(t), y(0)=0, y'(0)=-2 f(t)= t for 0≤t<1 and 0 for t≥1 Homework Equations The Attempt at a Solution L{y''}+6L{y}=L{t}-L{tμ(t-1)} where μ(t-1) is Unit Step Y(s)=L{y} sY(s)-y(0)=L{y'} and y(0)=0 s2Y(s)-sy(0)-y'(0)+6Y(s) where y(0)=0 and...

Homework Statement ##\mathcal{L}^{-1}\Big\{\frac{s}{(s^2+1)^2}\Big\}## I'm trying to figure out how to find the inverse Laplace transform of this expression. Is this something you just look up in a table or is there a way to find it directly, maybe by Convolution?

Hello, We were introduced to Laplace transforms in my ODE class a few days ago, so naturally I went online to try to figure out what this transform actually is, rather than being satisfied with being able to compute simple Laplace transforms. The Wikipedia page for the transform says it...

Homework Statement Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph). 1) f(t)=t Homework Equations The Attempt at a Solution Seems pretty easy... Question is, I don't understand the directions exactly.. Am I...

Homework Statement What is the laplace transform of H(-t-17) Homework Equations Shifting theorem: L(H(t-a)) = (e^-as)/s The Attempt at a Solution This is the only part of the problem that I can not get (this part is from a larger differential equation I'm trying to solve). I'm can't seem...

Find the Inverse Laplace of 1/(s^3) is there some special rule for cube? The answer is t^2/2 Looking at the Laplace Table t^n looks similar but its not it exactly. What should I do?

Homework Statement Determine the Laplace Transform of ∫(from 0 to t) (t-τ)cos(2(t-τ))e-4τ dτ using Laplace Transform tables. Homework Equations I know the basic convolution theorem is (f*g)(t) = ∫f(τ)g(t-τ)dτ The Attempt at a Solution I'm not sure if this is double convolution...

Hi. We are learning about Laplace transforms at uni and I must say that this is a real pain. I have one questions concerning the concept of Laplace transforms, and also a question concerning a specific transform. The task is to make a Laplace transform of: t*sin(2t). I could do an integration by...

Hey guys , having a bit of bother getting a solution for this question. Any help would be greatly appreciated! There is a Picture attached showing how far I have got ..

Homework Statement find the range of values for s: f(t)=e^(-t/2)u(t)Homework Equations The Attempt at a Solution what I did initially was to take the integral from 0 to infinity of e^(-st)e^(-t/2)u(t) dt this gave me 2/(2s+1) which when I sub in -.5 to find a divide by 0 or discontinuity then...

Please refer to the attachment. For part a) so far I have: $e^x = 1 + \frac{x}{1!} + ...+ \frac{x^n}{n!}$ So $S^\frac{-1}{2}e^\frac{-1}{S} = S^\frac{-1}{2}(1 -\frac{1}{S} + \frac{1}{2!S^2} - \frac{1}{3!S^3} + \frac{1}{4!S^4} + ... - ...)$ I don't think my $S^\frac{-1}{2}$ on the outside is...

Homework Statement Use Laplace transforms to derive an expression for the current flowing in the circuit shown in the figure, given that i = 0 when t=0 Homework Equations Expression for the current in an LR series circuit I(t) = V / R (1-e-Rt/L) V - Volts (6V) R - Ohms (10Ω) L...

Homework Statement A capacitor of 0.1 F and a resistor of 5 Ω are connected in series; the combination is applied to a step voltage of 20V. Determine the expression for the: (a) current that flows in the circuit and (b) the voltage across the capacitor in time domain. Homework Equations...

How do you find the laplace transform of this without expanding it? $ L(t^2+1)^2 $

The domain is \(0\leq x < \infty\) and \(0\leq y\leq b\). \begin{align*} \phi(x, 0) &= f(x)\\ \phi(x, b) &= 0 \end{align*} I want to show that \[ \phi(x, y) = \frac{1}{\pi}\int_0^{\infty} \int_{-\infty}^{\infty}f(\xi) \frac{\sinh[u(b - y)]}{\sinh(ub)} \cos[u(\xi - x)]d\xi du \] Why isn't the...

Please ignore this thread.

Homework Statement $$L\{ { e }^{ -t }*{ e }^{ t }cost\}$$ Homework Equations The Attempt at a Solution $$L\{ { e }^{ -t }*{ e }^{ t }cost\} \\ =L\{ \int _{ 0 }^{ t }{ { e }^{ -\tau }{ e }^{ t-\tau }cos(t-\tau )d\tau } \} \\ =\frac { L\{ { e }^{ t }cost\} }{ s } \\ =\frac {...

Homework Statement $${ { L } }^{ -1 }\{ \frac { s }{ { ({ s }^{ 2 }+1) }^{ 2 } } \} +{ { L } }^{ -1 }\{ \frac { 1 }{ { ({ s }^{ 2 }+1) }^{ 2 } } \}$$ Homework Equations The Attempt at a Solution I used ##{ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s...

Homework Statement I have some problem finding the inverse laplace transformation of the function: \frac{s}{s^2+2s+5} Homework Equations http://math.fullerton.edu/mathews/c2003/laplacetransform/LaplaceTransformMod/Images/Table.12.2.jpg The Attempt at a Solution I tried to...

\nabla^2 u=\frac {\partial ^2 u}{\partial x^2}+\frac {\partial ^2 u}{\partial y^2}=\frac {\partial ^2 u}{\partial r^2}+\frac{1}{r}\frac {\partial u}{\partial r}+\frac{1}{r^2}\frac {\partial ^2 u}{\partial \theta^2} I want to verify ##u=u(r,\theta)##, not ##u(x,y)## Because for ##u(x,y)##, it...

Homework Statement I'm stuck trying to find out the inverse Laplace of F(s) to get y(t) (the solution for the differential equation): Y(s) = 1 / [ (s-1)^2 + 1 ]^2 The Attempt at a Solution I tried using a translation theorem and then apply the sine formula, but the denominator...

Hey guys, i have read many posts on physics forums but this would be my first post. I am quite stuck so any help would be much appreciated. Homework Statement Use Laplace transforms to solve the initial value problem: f''(y) + 4f'(y) +8y = u(t-1) where y(0) = 1 and y'(0) = -1 Solve...

Homework Statement Can people help me on these two questions, please. Q1) Does f(t) have a Laplace transform F(s) for sufficiently large real value of s, where f(t) = et/(t4-1). Q2) Either find a function f(t) for which F(s) = L{f(t);t→s} = es, or explain why no such function f(t)...

In my differential equations book (Edward and Penny) there are many examples of Laplace transforms being applied to linear differential equations with constant coefficients and no examples of them being applied to linear differential equations with variable coefficients. My question is, can this...

Homework Statement I need to find the laplace transformation of the following function (and it's ok to leave it expressed as an integral). After doing the initial steps and algebra I got Y(s)= g(t)/(s+2)^2 + 7(1/(s+2)^2)+ 2(1/(s+2)^2) the answer is y(t)=2e^-2t +te^-2t...