Laplace Definition and 1000 Threads

Homework Statement The switch in the circuit in has been closed for a long time. At the switch is opened. a) Find i0 for t>=0 (b) find v0 for t>=0 I don't understand the effect of the switch in the position it is in. It's acting as a short circuit, I think, so presumably the capacitor doesn't...

Homework Statement The stability of a spinning body may be explored by using equation (3.40), with no torque components present. It will be assumed here that the spin is about the z -axis and has a rate ωZ = S. Homework Equations $$I_{xx}\dot{ω} - (I_{yy}-I_{zz})Sω_y = 0$$ $$I_{yy}\dot{y} -...

Homework Statement Y=(8s-4)/(s²-4) Homework EquationsThe Attempt at a Solution I rearranged the right side as: 8*(s/(s²-2²))-2*(2/(s²-2²)) Using the Laplace transform chart given in the class I was able to identify these as the transforms of hyperbolic sine and hyperbolic cosine making the...

Homework Statement Homework EquationsThe Attempt at a Solution 1. I got Y(s) = (15s +18)/(5s^2+s-2) 2. I got Y(s) = (7s - 7iw + 1)/((s+4)(5 - iw)) Was just wanting to make sure I solved these right. I would type it out but without formating, it will look messy.

Homework Statement Solve ##\frac{dy}{dt} -y = 1, y(0) = 0## using the laplace transform Homework EquationsThe Attempt at a Solution ##\mathcal{L}\big\{\frac{dy}{dt}\big\} - \mathcal{L}\big\{y\big\} = \mathcal{L}\big\{1\big\}## ##sY(s) - y(0) - \frac{1}{s^2} = \frac{1}{s}## ##Y(s)...

Homework Statement Solve the Laplace equation in 2D by the method of separation of variables. The problem is to determine the potential in a long, square, hollow tube, where four walls have different potential. The boundary conditions are as follows: V(x=0, y) = 0 V(x=L, y) = 0 V(x, y=0) = 0...

I have two functions ##\phi(t)=\cos(\omega t)## and ##f(t)=u(t)−u(t−k)## with ##f(t)=f(t+T)##, ##u(t)## is the unit step function. The problem is to find Laplace transform of ##\phi(t) \cdot f(t)##. I have tried convolution in frequency domain, but unable to solve it because of gamma functions...

Hello, I am studying control theory. And I have encountered something I have never considered or thought about. Consider a system with y as the output differential equation and u as the input. any(n) + ... + a1y(1) + a0y = bmu(m) + ... + b1u(1) + b0u Here, the subscripts indicate...

Hello, I have begun to teach myself Control Theory. I am looking for a book that is focused for mechanical engineers. I do not mind examples in electrical engineering, but they bore me (no offense). Also, I find some books begin with Laplace Transforms. Yet I found this online lecture...

Homework Statement Prove that ##\vec {a} \cdot (\vec {b} \wedge \vec {C_r}) = \vec {a} \cdot \vec {b} \vec {C_r} - \vec {b} \wedge (\vec {a} \cdot \vec {C_r})##. Note that ##\vec {a}## is a vector, ##\vec {b}## is a vector, and ##\vec {C_r}## is an r-blade with ##r > 0##. Also, the dot...

Homework Statement I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...

Homework Statement I am given this equation: and asked to solve using Laplace transforms The Attempt at a Solution This is what I did: This seemed logical to me, I used partial fractions and it stayed pretty simple. This is what the solutions my prof posted do: Is my answer equivalent...

Homework Statement 2. Homework Equations [/B] I'm studying a mathematical behaviour of a servo motor and I need some help to understand it. The output signal is \$\beta (t)\$, representing the angle rotated by the axis at instant t, in relation to the equilibrium position. On the servomotor...

Dear all, I would need mathematical help to solve for the temperature field in an annular geometry (you find a picture attached below the text): A copper pipe containing a boiling two-phase flow (in the stratified regime) is immersed in a liquid bath, which temperature ##T_{IY}## is assumed to...

Homework Statement Homework Equations If i solve the wave equation using separation of variable and laplace tranform. Will i get the same answer ? The Attempt at a Solution

Homework Statement Homework EquationsThe Attempt at a Solution I'm kind of lost now, how do I go about getting this into the right form for partial fraction exapnsion. And also what do I do with the V(0'). There was no information given about it.

Homework Statement Hi, So I had a pretty long question solving a Linear ODE but now I've gotten stuck at this stage where I can't seem to get it into the right form to carry out partial fraction expansion Homework EquationsThe Attempt at a Solution [/B] I'm quite sure that I what I have at...

Hi, I need to solve Laplace equation ## \nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ## r_1<r<r_2 ##, ## 0<z<L ##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...

Homework Statement 4(d2x/dt2) +3x = t*e-3tsin(5t) Homework EquationsThe Attempt at a Solution So I know how to take the Laplace transform and find the function for the Laplace domain: X(s) = 10(s+3)/(((s+3)2+25)2)(4s2+3) + (10s/(4s2+3)) + (2/(4s2+3)) But trying to convert...

Hello, I have the following PDE equation: a*b/U(u)*V(v) = 0 where a and b are arbitrary constants, and U an V are two unknown functions. To me it appears this has no solution, however I would like to ask if anyone has some suggestions, such as transforming it to another type using Fourier or...

Homework Statement am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it? appreciated, Homework EquationsThe Attempt at a Solution my attempt is the same as in the attached picture...

A harmonic function is one that satisfies Laplace's equation -- a definition cannot be more precise than that. However, in the study of vibrations, sine and cosine are considered harmonic functions; but they don't solve Laplace's equation. And then there are words like: harmonics (for higher...

Homework Statement L-1{[(2s-1)]/[(s^2)(s+1)^3]} Homework Equations L{f(t)e^(at)}=F(s-a) The Attempt at a Solution I have tried million ways but the different exponents in the denominator are throwing me off. The other problem is that I cannot use partial fractions, the homework instructions...

Homework Statement You have to calculated the Laplace transformation for 1/ cos(t) Homework Equations That's all The Attempt at a Solution i tryed whit some trigonometric formulas but i don't get anywhere : 1/cos(t) = cos(t) / (1- sin ^2 (t)) or 1/cos(t) = cos(t) + sin(t) x tg(t) or...

Hi, I need to solve Laplace equation ##\nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ##r_1<r<r_2##, ##0<z<L##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...

Hi, I need to solve Laplace equation:## \nabla ^2 \Phi(x,r)=0 ## in cylindrical domain ##0<r<r_0##, ##0<x<L## and ##0<\phi<2\pi##. The boundary conditions are the following ones: ## \left\{ \begin{aligned} &C_{di}\Phi(x,r_0)=\epsilon \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_0} \\...

Hi. I have this problem in trying to solve this PDE analytically. The PDE is represented by this diagram: Basically this is solving the Laplace equation with those insulated boundaries except it has that point diffusing its value across the plane. I know how to solve the Laplace equation...

Homework Statement A step voltage of 120v is applied to a series CR circuit. R = 20KΩ, C = 4µF 1. Deduce, using Kirchoff's voltage law and Laplace Transforms, an expression for the transient circuit current. 2. Using the equation obtained in 1. deduce the equations for the transient voltages...

Homework Statement I have to take the inverse Laplace of this function (xoms+bxo)/(ms2+bs+k) this can not be broken into partial fractions because it just gives me the same thing I started with. How is this done? This is coming from the laplace of the position function for a harmonic oscillator...

Prelude Consider the convolution h(t) of two function f(t) and g(t): $$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$ then we know that by the properties of convolution $$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$ Intermezzo We also know that...

Homework Statement The Attempt at a Solution At this point, usually I would replace the values and sometimes separate into partial fractions, and then take the inverse Laplace transformation. So I know that the inverse Laplace needs to give me 6+12e^-2t. I am given the answers in my...

Homework Statement The Attempt at a Solution As you can see, the answer is given in the problem. I have the 0.8t par right but somewhere along the way I messed up. I'm not really to sure about how to solve these types of problems using Laplace. Is there a general procedure that I should be...

Homework Statement A beam is supported at one end, as shown in the diagram (PROBLEM 11 page 281 of Lea, 159 of the course pack). A block of mass M and length l is placed on the beam, as shown. Write down the known conditions at x = 0. Use the Laplace transform to solve for the beam...

Homework Statement "Suppose that ##F(s) = L[f(t)]## exists for ##s > a ≥ 0##. (a) Show that if c is a positive constant, then ##L[f(ct)]=\frac{1}{c}F(\frac{s}{c})## Homework Equations ##L[f(t)]=\int_0^\infty f(t)e^{-st}dt## The Attempt at a Solution ##L[f(ct)]=\int_0^\infty f(ct)e^{-st}dt##...

I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...

This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...

I'm having trouble solving the differential equation (a-bx)y'+(c-dx)y-e=0 with a,b,c,d,e constants. I tried laplace transforming it, but then I end up with yet another differential equation in the laplace domain because of the xy and xy' terms.

Homework Statement ƒ(s) = 1/((1-exp(-s))*(1+s)) Homework EquationsThe Attempt at a Solution I know the solution is periodic but how to obtain the t-domain function?

Hi I have a series ${f}_{1}$ , ${f}_{2}$, ... that are all a functions of a variable $t$ I am seeking a point-wise convergence. to investigate the convergence of the series I took Laplace transform. If I can find a condition on the Laplace variable $s$, can I find the condition of convergence...

Homework Statement ty'' + y' = 2t2, y(0) = 0Homework Equations laplace(f''(t)) = s2laplace(f(t)) -sf(0) - f'(0) (-1) (d/ds) (F(s))The Attempt at a Solution I know how to solve the problem except for the ty'' part. I tried using the equation and I got -d/ds(s2Y(s) - 0 - f'(0)) which becomes...

Homework Statement Solve Laplace Transform L{tsin(2t)sin(5t)} Homework Equations cos(bt)=s/s^2+b^2 trig identity (product identity): sin(a)sin(b)=1/2[cos(a-b)-cos(a+b) t^nf(t)=(-1)^nd^n/ds^nF(S) (the template is complicated for me to use. Srry for the inconvinience) The Attempt at a Solution...

Homework Statement Determine the inverse Laplace transform Homework Equations 3s+9. (s+3)^2+7 The Attempt at a Solution [/B] Hi iam new to the forum and still unsure how to make the equations the correct format so hope you can understand what I have typed. I have Tried to Convert...

Homework Statement Determine the inverse Laplace transform Homework Equations 3s+9/(s+3)^2+7 The Attempt at a Solution Converted to 3s+9/s^2+6s+16 to try and use the partial fractions method but getting nowhere. I'm Not sure if Iam making the question more difficult, can't seem to put the...

Homework Statement Deriving an s-domain equation for the following inputs a) &b) The Attempt at a Solution I understand how to derive the equation for an input with zero initial conditions (part a) but I'm not sure what to do when there are non-zero initial conditions (part b)

Homework Statement I want to invert a function from Laplace transform space to normal space. Homework Equations In Laplace transform space, the function takes the form $$ \bar f (s) = \frac{\exp\left[ x (-a +\sqrt{a^2+ b +c s} )\right]}{-a +\sqrt{a^2+ b +c s}}. $$ Here, ##s## is the Laplace...

Hi. I read this thread with great interest and have similar question: In a deterministic universe, does entropy exist for Laplace's demon? Since he knows the universe to it's microstate, does the term "macrostate" even make sense to him? And say there is a "half-demon" that only knows the...

Hi Guys, I have an equation I'm trying to work out if it 'Laplaceable'? As in, can I get a transfer function? d2y(t)/dt2 = k⋅cos(Φ)⋅y(t)

Homework Statement Homework Equations N/A The Attempt at a Solution The left hand side (red box) is the data sheet provided to us in the exam. The right hand side (blue box) is Wolfram Alpha. The data sheet says that the inverse Laplace transform of 1/s is equal to u(t) (i.e. the unit step)...

Consider an infinitely long hollow dielectric cylinder of radius a with the electricpotential V = V0 cos φ on the surface of the cylinder where φ is an angle measured around the axis of the cylinder. Solve Laplace’s equation to find the electric potential everywhere in space.Do you just plug V...

Homework Statement Homework EquationsThe Attempt at a Solution 1 ##3u(t)-3u(t-2)## ##\frac{3}{s}-\frac{e^{-2s}3}{s}## 2 ##\frac{5t}{2}u(t)-\frac{5t}{2}u(t-2)## ##\frac{5}{2s^2}-\frac{5e^{-2s}}{2s^2}## 3 I have no idea how to do this one.