What is Laplace transforms: Definition and 186 Discussions

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable

t

{\displaystyle t}
(often time) to a function of a complex variable

s

{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral

L

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{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}

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1. I On a bijective Laplace transform

Exponential polynomials are linear combinations of terms like ##t^ne^{at}\cos{bt}## and ##t^ne^{at}\sin{bt}##, where ##n## is a nonnegative integer, ##a## is real and ##b>0##. Like the proper rational functions, these are presumably subspaces of ##\mathbb R^{\mathbb R}##. The proof goes like...
2. What is the best way to introduce Laplace transforms for Engineers?

Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function. How can Laplace transforms be introduced so that students are...
3. MHB What is the best way to introduce Laplace transforms in an Engineering Mathematics course?

Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function. How can Laplace transforms be introduced so that students are...
4. Laplace transforms for which value of s?

I was wondering how you work out what values of s a Laplace transform exists? And what it actually means? The example given in class is an easy one and asks to calculate the Laplace transform of 3, = 3 * Laplace transform of 1 = 3 * 1/s. Showing this via the definition, where does the range of s...
5. C

Mathematical Engineer, Electrical Engineer, & Author

My History -------------- I attended Oregon State U. and majored 3 years in Electrical Engineering. Then I switched to a Math major for my final years and graduated with a B.S. in Math (1967). Developed several Apps for Engineers & Scientists. My first job was with Lockheed Aircraft Co...
6. Engineering Laplace transform of the given circuit

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?

13. MHB Hello i would like some help with laplace transforms.

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...
14. MHB Solving PDE using laplace transforms

[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 $x$...
15. Engineering RLC circuit solved with Laplace transformation

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...
16. Laplace & Inverse Laplace transforms

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...
17. Engineering Laplace tranforms, transient current series CR circuit

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...
18. Laplace transforms in circuit analysis, finding missing values

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...
19. A tricky inverse Laplace transform

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...
20. Engineering Circuit Analysis with Laplace Transforms

Homework Statement Homework Equations V=IR All of them actually The Attempt at a Solution So I Started off by transforming the voltage source into the 's' domain vs(s) = (4/s) -(4/s)*e-.5t I know the initial conditions are zero, in other words at t=0, the voltage and currents at the...

33. Using Laplace transforms to solve differential equations - with a twist

I've been given this: x''+ x = 4δ(t-2π) The question asks: With initial conditions of x(0) = 1 and x'(0) = 0, find x(t) using Laplace transforms. I can easily get it to this: 4(sin(t-2π)u(t-2π)) But the question says "express your final solution without use of the unit step function". This is...
34. MHB Laplace Transforms (proofs of)

I should state, from the outset, that this tutorial is NOT going to go into any great detail about the theory and applications of Laplace transforms. Some of the aforementioned will be discussed in a cursory way, but the aim here is merely to provide a selection of proofs for common transforms...
35. Inverse Laplace Transforms Problem 2

Homework Statement f(s) = 6/s^2-9 Homework Equations I think f(t) = (1/b-a)(e^-at-e^-bt) The Attempt at a Solution Replace 6/s^2-9 with 6/(s-3)(s+3) a=-3 b=3 Plug in (1(6)/3-(-3))(e^-(-3)t-e^-3t) Final Result e^3t-e^-3t
36. Applying Inverse Laplace Transforms to f(s) = -5s/S^2+9

Homework Statement f(s) = -5s/S^2+9 Homework Equations I think f(t) cosωt = f(s) s/s^2+ω^2 The Attempt at a Solution ω=3 Answer -5cos(3t) Can anyone tell me if I did this correctly? I think I did but just want to make sure, if not can you tell me what I did wrong? Thanks
37. Problems with Laplace Transforms

Homework Statement The coordinates ##(x,y)## of a particle moving along a plane curve at any time t, are given by \frac{dy}{dt} + 2x=\sin 2t, \frac{dx}{dt} - 2y=\cos 2t. If at ##t=0##, ##x=1## and ##y=0##, using Lapace transform show that the particle moves along the curve 4x^2+4xy+5y^2=4...

47. Laplace transforms to solve initial value DE / partial fractions

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...
48. Using laplace transforms to solve integrals

Homework Statement ##\int_0^\infty \frac{a}{a^2+x^2} dx## Homework Equations All the basic integration techniques. The Attempt at a Solution So, I saw this problem and wanted to try it using a different method then substitution, which can obviously solve it pretty easy. Since it is a very...
49. Optimal Controller Gain for Desired Response in 2nd Order Laplace Transforms

Homework Statement Solve the DE for y(t) with the IC's y(0)=20.8m/s and y'(0)=0 if the input is a step function scaled by the desired velocity Vo. vd(t)=Vou(t). Assume the desired velocity Vo=27.8m/s Homework Equations y''(t) + (D/M)y'(t) + (K/M)y(t) = (K/M)vd(t) M = 1,000kg D = 100kg/s K...
50. MHB ?'s question at Yahoo Answers regarding Laplace Transforms

Here is the question. Here is a link to the question: Can you show the steps of this differential equations problem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.