Laplace transform Definition and 779 Threads

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...

Hey all, I hope this is the correct forum section to post this in. I heard about this problem from a youtube video but I've not been able to simulate it because the video was meant only for an introduction into PID control. Here's the problem: A remote control helicopter is hovering just...

Homework Statement Solve the following partial differential equation , using Fourier Transform: Given the following: And a initial condition: Homework EquationsThe Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...

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...

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...

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}...

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?

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)...

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...

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 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 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...

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 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 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 am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem. A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...

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...

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 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 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)...

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.

Homework Statement Can someone check my work? Homework EquationsThe Attempt at a Solution 1. ##\frac{1}{s+2}+\frac{1}{s^2+1}## 2. ##\frac{2}{s}+\frac{3}{s+4}## 3. ##\frac{s*sin(-2)+cos(-2)}{s^2+1}## 4. ##\frac{1}{(s+1)^2}## 5. Don't really know how to do this one...

Hi, everyone, the question is as below: Find the inverse Laplace transform to 1/(350+s) * X(s). 's' is the Laplace variable and 'X(s)' is also a variable. I inverted 1/(350+s) and X(s) separately and multiplied them together directly. But this seems not giving me the correct answer. Could...

Not homework question, just need clarification and explanation. How did the person get from the left equation to the right side. I know he's just simplifying. But he didn't include steps and I've been trying to work out how to no avail. Any help on how this person simplified the LHS to RHS? Thanks!

Homework Statement Solve: y''+λ^2y = cos(λt), y(0) = 1, y'(π/λ) = 1 where t > 0 Homework EquationsThe Attempt at a Solution I start off by taking the Laplace transform of both sides. I get: L(y) = \frac{s}{(s^2+λ^2)^2}+\frac{sy(0)}{s^2+λ^2}+\frac{y'(0)} {s^2+λ^2} Now take the inverse...

Homework Statement find the inverse Laplace transform of the given function by using the convolution theorem Homework Equations F(s) = s/((s+1)(s2)+4) The theorem : Lap{(f*g)(t)} = F(s)*G(s) The Attempt at a Solution I know how to find it the answer is : we have 1/(s+1) * s/(s+4) and the...

Homework Statement We want to find the Laplace transform for f(t): 0 for t≤2 and (t-2)2 for t≥2 Homework Equations I know that Lap{uc f(t-c)} = e-csLap{f(t)}=e-csF(s) I rewrite f(t)=0+g(t) where g(t) = 0 for 0≤t<2 and (t-2)2 for t≥2 so we can write f(t)=g(t)= u2(t)*(t-2)2...

Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...

Homework Statement ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0##Homework Equations ##\int u'v = uv - \int uv'## The Attempt at a Solution This is a system of linear DE: ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0## Where the constants A-D are non-zero and x and y are functions of time. This is...

As the Heaviside function is a function of t - 4, that means all other terms must also be functions of t - 4. The sine function is, but the exponential isn't. However with a little manipulation, we get $\displaystyle \begin{align*} f\left( t\right) &= \mathrm{H}\,\left( t - 4 \right) \,\sin{...

It's not entirely obvious what to do with this question, as the denominator does not easily factorise. However, if we realize that $\displaystyle \begin{align*} s^4 + 40\,000 = \left( s^2 \right) ^2 + 200^2 \end{align*}$ it's possible to do a sneaky completion of the square... $\displaystyle...

Hello. I am reviewing the use of the Laplace Transform to do circuit analysis and I am slightly confused about the transform of a constant voltage source. For example, let's say we have a constant voltage source V1(t) applied to a circuit for a long time - let's say it reaches steady state. We...

Use laplace Transform to solve this ode: So I got: sV(s)-V(0)-12V(s)=U(s+5) V(s)(s-12)=U(s+5)+1 V(s)=[U(s+5)+1]/(s-12) Now to go back to time domain with Inverse Laplace Transform...My question is, how to transform U(s+5)/(s-12)? Any help? Thanks guys

Homework Statement FIGURE 4(a) represents a system to measure acceleration (i.e. an accelerometer). It shows a piezoelectric crystal that is connected to an amplifier and display via a length of coaxial cable2.A piezoelectric current is produced when the crystal is distorted by an applied...

Hi members, Laplace transform using differential equations.(see attached PDF file) My question d/ds(s^2 y- s Y(0)-Y'(0).)... Y(t)=sin(sqrt(t)) Y(o)=0 Now Y'= cos(sqrt(t)/2sqrt(t) Y'(0)=infinity d/ds (Y'(0)=?? can it be treated as a constant or can we change limit and differentiation??I...

Homework Statement find the laplace transform of (e^-s) / [ (s)(s-3) ] since there's (e^-s) which can be found in L { f(t-a) H(t-a) } = (e^-(as)) F(s) , so , i found a = 1 , then i found F(s) = 1/ [ (s)(s-3) ] , formula : i have attached the working below , is it correct ? btw , the...

Homework Statement Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. Then solve the differential equation with...