Laplace Definition and 1000 Threads

Hello i have an assignment. From given circuit i need to find s domain and inverse them back to t domain. can you help me by explain this circuit?

i want to ask about my homework im not understand what to do with this formula :

I used partial fraction method first as: 1/s(s^2+w^2)=A/s+Bs+C/(s^2+w^2) I found A=1/w^2 B=-1 C=0 1/s(s^2+w^2)=1/sw^2- s/s^2 +w^2 Taking invers laplace i get 1/w2 - coswt But the ans is not correct kindly help.

Since this is of the form $\displaystyle \frac{f\left( t \right)}{t} $ we should use $\displaystyle \mathcal{L}\,\left\{ \frac{f\left( t \right) }{t} \right\} = \int_s^{\infty}{F\left( u \right) \,\mathrm{d}u } $. Here $\displaystyle f\left( t \right) = \cosh{\left( 4\,t \right) } - 1 $ and so...

Take the Laplace Transform of the equation: $\displaystyle \begin{align*} s\,Y\left( s \right) - y\left( 0 \right) + 11\,Y\left( s \right) &= \frac{3}{s^2} \\ s\,Y\left( s \right) - 5 + 11\,Y\left( s \right) &= \frac{3}{s^2} \\ \left( s + 11 \right) Y\left( s \right) &= \frac{3}{s^2} + 5 \\...

Upon taking the Laplace Transform of the equation we have $\displaystyle \begin{align*} s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 4\,Y\left( s \right) &= -\frac{8\,\mathrm{e}^{-6\,s}}{s} \\ s^2 \,Y\left( s \right) - 2\,s - 0 + 4\,Y\left( s \right) &=...

Taking the Laplace Transform of the equation gives $\displaystyle \begin{align*} s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) - 5\left[ s\,Y\left( s \right) - y\left( 0 \right) \right] - 6\,Y\left( s \right) &= -\frac{126\,\mathrm{e}^{-6\,s}}{s} \\ s^2\,Y\left( s \right) -...

This requires the convolution theorem: $\displaystyle \int_0^t{f\left( u \right) \,g\left( t- u \right) \,\mathrm{d}u } = F\left( s \right) \,G\left( s \right) $ In this case, $\displaystyle g\left( t - u \right) = \mathrm{e}^{-3\,\left( t - u \right) } \implies g\left( t \right) =...

Start by taking the Laplace Transform of both equations, which gives $\displaystyle \begin{cases} s\,X\left( s \right) - s\,x\left( 0 \right) + X\left( s \right) + 6\,Y\left( s \right) = \frac{6}{s} \\ s\,Y\left( s \right) - s\,y\left( 0 \right) + 9\,X\left( s \right) + Y\left( s \right) = 0...

The Heaviside function suggests a second shift, but to do that, the entire function needs to be a function of $\displaystyle t - 4$. Let $\displaystyle u = t - 4 \implies t = u + 4$, then $\displaystyle \begin{align*} \mathrm{e}^{5\,t} &= \mathrm{e}^{5\left( u + 4 \right) } \\ &=...

hello if someone could please tell me if i am incorrect and where , and how to type it into a math program so it can understand it many thanks stephan2124 L -3e^{9t}+9 sin(9t) L-3e^{9t}+L 9 sin (9t) -3 Le^{9t}+9 L sin(9t) -3 (1/s-9) +9 (9/(s^2+9^2)) -3 (1/s-9) +9 (9/(s^2+81)) into a math...

f(t)=-5t^2+9\int_{0}^{t} \,f(t-u)sin(9u)du

$\displaystyle \begin{align*} \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \sinh{ \left( 11\,t \right) } \right\} &= \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \cdot \frac{1}{2} \left( \mathrm{e}^{11\,t} - \mathrm{e}^{-11\,t} \right) \right\} \\ &= \frac{5}{2} \,\mathcal{L} \left\{...

So I have completed (a) as this (original on the left): I have then went onto (b) and I have equated T(s)=Z(s) as follows: and due to hence Does this look correct to you smarter people? Thanks in advance! All replies are welcome :)

Laplace pointed out that the variation in pressure happens continuously and quickly. As it happens quickly, there is no time for heat exchange. This makes it adiabatic. But Newton believed it to be isothermal. Why isn't it isothermal but adiabatic? Why is there a change in temperature?

I have found various formulations for the Laplacian and I want to check that they are all really the same. Two are from Wikipedia and the third is from Sean Carroll. They are: A Wikipedia formula in ##n## dimensions: \begin{align} \nabla^2=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial...

Problem: Find a (limited?) solution to the diff eq. At the end of the solution, when you transform $$\frac{-1}{s+1} + \frac{2}{s-3}$$ why doesn't it become $$-e^{-t} + 2e^{3t} $$, t>0 ?

So the task is to solve the following integral with laplace transform. Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)). What I am unsure about is what happens with the integral part and how do we inpret the resulting expression? What will it result...

The circuit to be analyzed is shown below: Since initial conditions are zero (from the instructions) I will use laplace transforms for the cirucit and I will use the MAME method to solve this circuit. The laplace transforms that are required will give me: $$E_g(s) = \frac{10}{s}$$ $$ L_3 =...

We first determine the Laplace force for each value ##\alpha##. $$F_{\alpha} = 5(0.3)(0.4)\sin(\alpha) = 0.6\sin(\alpha) \ \text{N}$$ We then calculate the torque at angle ##\alpha##. $$\tau_{\alpha} = \frac{a}{2} F_{\alpha} = 2.5 F_{\alpha} \ \text{N.m}$$ Then we just plug in ##\alpha## and...

[Solved] Solving PDE using laplace transforms Hey, I'm stuck on this problem and I don't seem to be making any headway. I took the Laplace transform with respect to t, and ended up with the following ODE: $\frac{\partial^2 W}{\partial x^2}-W(s^2+2s+1)=0$ and the boundry conditions for $x$...

e

Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...

Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...

My solution is in the file shown here

So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd...

Imagine to be in 2 dimensions and you have to find the potential generated by 4 point-charges of equal charge located at the four corners of a square. To do that I think we simply add all the contributions of each single charge: $$V_i(x, y) = - \frac k {| \mathbf r - \mathbf r_i|}$$ $$ V(x, y)...

In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as... $${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$ The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...

I have used Laplace transform during my EE studies to solve differential equations and in control system analysis, but we were taught that as a tool kit to make the math easier. The physical meaning was never explained. I know basic time and frequency domain concepts (thanks to Fourier series)...

Vo(S) = [ N(s)Vi(s) + (- s2 + s - 2) ] / s3 + s2 + 1 ; can ignore (-s^2 + s - 2). From relevant equations: Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d3Vo(t)/dt3) + (d2Vo(t)/dt2) + Vo(t) = N(t)(dvi)/dt L[vi(t)] = t to s domain: [s3Vo(s) - s2Vo(0-) - SV'o(0-) - Vo''(0-)]Vo(s) + s2 - SVo -...

Hello, guys. I'm currently working on a physics problem that requires me to evaluate the inverse Laplace of the function in the attached file. When b = 0, "y" vanishes, and all one has to do is to look up the Laplace table for the inverse. However, non-zero b has been giving me a headache. I...

Homework Statement I have a value of $$ U=U_0+x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)+1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+...$$ We need to find the mean value of the U. So the answer is $$\overline{\rm U}\approx U_0+a^2/24(∇^2U)$$Homework Equations $$\overline{\rm U}=1/a^3 \int \int\int Udxdydz$$ The...

Homework Statement L{sin(ωt)/[1+cos^2(ωt)]} = Homework Equations d {arctan[cos(ωt)]} /dt = - ω•sin(ωt)/[1+cos^2(ωt)] The Attempt at a Solution ∫e^(-st)•[sin(ωt)/(1+cos²(ωt)] dt = -(1/ω)•∫e^(-st)•{arctan[cos(ωt)]}' dt = = (integrating by parts and taking Re(s) > 0) = = π/(4ω) -(s/ω)•∫...

I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier

Homework Statement I have to find the L-transform of ##f(x) = cos(\omega t + \phi)## Homework Equations . The Attempt at a Solution The straightforward approach is to write ##cos(\omega t + \phi)## as ##cos(\omega t)cos(\phi) - sin(\omega t)sin(\phi)## and it becomes: $$Lf(s) = \frac {s...

Homework Statement There's a metal cunducting cube with edge length ##a##. Three of its walls: ##x=y=z=0## are grounded and the other three walls: ##x=y=z=a## are held at a constant potential ##\phi_{0}## . Find potential inside the cube. Homework Equations The potential must satisfy Laplace...

Hi PF! I looked through the documentation on their website, but under the tab "Solve partial differential equations over arbitrarily shaped regions" I am redirected to a page that does not specify how to create a region. Any help is greatly appreciated. Also, if it helps, the domain is a...

can anyone help me on how I can map an isosceles trapezoid onto a rectangular/square domain.Actually I need to solve Laplace equation(delta u = 0) over this isosceles trapezoidal domain. Schwarz Christoffel mapping may help me. But can anyone give me any hint on this mapping procedure?

By using the laplace transform: $f(t)=sin(Φ(t))$ I want it in the form: F(S)/Φ(S) The purpose is to linearize it in order to put it into a larger transfer function, so far my only solution is to simplify it using taylor expansion.

What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of...

Homework Statement I’m being asked to prove if and why (what instances in which) T<0 for the Laplace transform property of time shifting doesn’t hold. Homework Equations L{f(t-T)}=e^-aT* F(s) The Attempt at a Solution I know that for T<0 there are instances where the property cannot hold, but...

Hello! (Wave) I want to solve the Laplace equation on the unit disk, with boundary data $u(\theta)=\cos{\theta}$ on the unit circle $\{ r=1, 0 \leq \theta<2 \pi\}$. I also want to prove that little oscillations of the above boundary data give little oscillations of the corresponding solution of...

Hi, I am looking for the solution to the quadrant problem of the Laplace equation in 2 dimensions with Dirichlet boundary conditions \begin{equation} \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \end{equation} in the first quadrant ## x, y \geq 0 ## with boundary...

Homework Statement The input signal of the circuit shown below is ##x(t)=2\sin (ω_ot + \pi/6)##. The switch in the circuit is controlled with a digital signal of the form ##s(t)=\sum_{k=-\infty}^{+\infty} (u(t+ε-kT_s) - u(t-ε-kT_s))##, ##\frac{2\pi}{T_s}=800\pi##, ##ε\to 0##, so that when the...

Hi, i need some help here. Can you help me?:sorry: Here is the problem. Exercise statement: The switch have been closed for a long time y is opened at t=0. Using Laplace's transtormation calculate V0(t) for t ≥ 0 This is what i made to solve it: 1) I know while the switch is closed, the...

Hello! (Wave) I want to find $f(t)$ if its Laplace transform is $F(s)=\frac{1}{s(s^2+1)}$. We use the following formula, right? $$f(t)=\frac{1}{2 \pi i} \lim_{T \to +\infty} \int_{a-iT}^{a+iT} e^{st} F(s) ds$$ But how can we calculate the integral $\int_{a-iT}^{a+iT} e^{st}...

Hello! (Wave) Let $a,b>0$ and $D$ the rectangle $(0,a) \times (0,b)$. We consider the boundary value problem in $D$ for the Laplace equation, with Dirichlet boundary conditions, $\left\{\begin{matrix} u_{xx}+u_{yy}=0 & \text{ in } D,\\ u=h & \text{ in } \partial{D}, \end{matrix}\right.$...

If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?