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


{
f
}
(
s
)
=

∫

0


∞


f
(
t
)

e

−
s
t



d
t
.


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

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  1. P

    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. matqkks

    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. matqkks

    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. H

    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. L

    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?
  7. P

    MHB Oscar's question via email about solving a DE using Laplace Transforms

    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) + 7 \left[ s\,Y\left( s \right) - y\left( 0\right) \right] + 6\,Y\left( s \right) &= \frac{60\,\mathrm{e}^{-6\,s}}{s} \\ s^2\,Y\left( s \right) -...
  8. P

    MHB Adam's question via email about Laplace Transforms

    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 \\...
  9. P

    MHB Massaad's question via email about Laplace Transforms

    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) -...
  10. P

    MHB Mahesh's question via email about Laplace Transforms (2)

    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) =...
  11. P

    MHB Mahesh's question via email about Laplace Transforms (1)

    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...
  12. P

    MHB Alexander's question via email about Laplace Transforms

    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) } \\ &=...
  13. S

    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. T

    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. MaxR2018

    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. Cocoleia

    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. D

    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. Cocoleia

    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. H

    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. gmm

    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...
  21. P

    MHB Collin's question via email about solving a DE using Laplace Transforms

    Taking the Laplace Transform of both sides we have $\displaystyle \begin{align*} \mathcal{L}\,\left\{ y'' + 4\,y \right\} &= \mathcal{L}\,\left\{ \mathrm{H}\,\left( t - 7 \right) \right\} \\ s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 4\,Y\left( s \right) &=...
  22. P

    MHB Collin's questions via email about Inverse Laplace Transforms

    As the denominator is a function of s + 3, it suggests a shift had to have been utilised. As such, we also need the numerator to be a function of s + 3... Let $\displaystyle \begin{align*} u = s + 3 \end{align*}$, then $\displaystyle \begin{align*} s = u-3 \end{align*}$ and thus...
  23. R

    Inverse Laplace Transform

    Homework Statement Given the Laplace transform $$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$ by using the complex inversion formula compute the inverse Laplace transform, ##f(t),## for the following regions of convergence: (i) ##Re(s)<-2;## (ii) ##-2<Re(s)<0;## (iii) ##Re(s)>0.## Homework Equations...
  24. S

    I What are the Laplace transforms of powers of y?

    Let's say you have a function y(t). You know how derivatives of y have their own Laplace transforms? Well I was wondering if powers of y such as y^2 or y^3 have their own unique Laplace transforms as well. If so , how do you calculate them (because plugging them into the usual integral doesn't...
  25. Aristotle

    Laplace Transform Method for Solving Initial Value Problems

    Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..) y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF - f(0) Lf = F(s) The...
  26. Aristotle

    Initial Value Problem using Laplace Transform help?

    Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..) y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF -...
  27. C

    Laplace Transforms: Transfer Functions and Impulse

    Homework Statement I uploaded the question as a picture and attached it. Homework Equations Unit step function - u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases} Impulse function - δ(t) = \displaystyle\lim_{Δ\rightarrow 0} δ_Δ (t) Multiplication Property...
  28. C

    Laplace Transforms: Transfer Functions, and IVT/FVT Problems

    Homework Statement I uploaded the problem statements as a picture as well. I have completed these and was wondering if someone could check my work, and let me know if it is correct. Problem 1.3: Find the expression for the transfer function of this linear time-invariant causal system with...
  29. C

    Laplace Transforms Involving: Unit-Step, and Ramp Functions

    Homework Statement Here is an imgur link to my assignment: http://imgur.com/N0l2Buk I also uploaded it as a picture and attached it to this post. Homework Equations u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases} The Attempt at a Solution Question 1.1 -...
  30. khurram usman

    A discussion about Fourier and Laplace transforms and calculus

    i have read many of the answers and explanations about the similarities and differences between laplace and Fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of laplace. Some signals have Fourier but laplace is not defined , for instance cosine or sine...
  31. J

    Laplace Transforms Homework: Initial Displacements & Velocities

    Homework Statement Homework Equations Laplace Transforms The Attempt at a Solution Using basic physics knowledge I got m1a1=-k1x1+k2(x2-x1) and m2a2=-k3x2-k2(x2-x1) Sub in values and use laplace transforms and rearrange partial fraction and I found that By doing this I am assuming...
  32. S

    MHB Yes, that is correct! Good job!

    Solev by Laplace Transforms $y'' - 5y' + 6y = 1$ $y(0) = 1$, $y'(0) = 0$So I am getting stuck. Here's my work $s^2Y - 5sY + 6Y = \frac{1}{s} + s - 5$ multiplied through by $s$ to get $s^3Y - 5s^2Y + 6sY = 1 + s^2 - 5s$ so: $Y = \frac{1+s^2-5s}{s^3-5s^2+6s}$ so: $1+s^2-5s = \frac{A}{s} +...
  33. L

    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. DreamWeaver

    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. S

    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. S

    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. F

    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...
  38. P

    MHB Douglas' questions about Laplace Transforms

    1. $\displaystyle \begin{align*} y''(t) + 7y'(t) + 12y(t) &= -36H(t - 2) \textrm{ with } y(0) = 0, y'(0) = 0 \\ \mathcal{L} \left\{ y''(t) + 7y'(t) + 12y(t) \right\} &= \mathcal{L} \left\{ -36H(t-2) \right\} \\ s^2\,Y(s) + s\,y(0) + y'(0) + 7 \left[ s\,Y(s) - y(0) \right] + 12Y(s) &=...
  39. vanceEE

    Derivative of Laplace Transforms

    Homework Statement $$xy'' + xy = 0$$Homework Equations y'(0) = 0, y(0) = 1The Attempt at a Solution $$ L[xy''] + L[xy] = 0$$ $$-L[-xy''] - L[-xy] = 0$$ $$-\frac{d}{dp}L[y''] - \frac{dY}{dp} = 0 $$ $$-\frac{d}{dp}(-y'(0)-py(0) + p^2Y) - \frac{dY}{dp} = 0 $$ $$\frac{d}{dp}(-p + p^2Y) +...
  40. P

    MHB Douglas' question regarding Laplace Transforms (1)

    Hi Douglas, I agree, partial fractions would be the best approach. Notice that $\displaystyle \begin{align*} s^4 + 1024 &= \left( s^2 \right) ^2 + 2\cdot s^2 \cdot 32 + \left( 32 \right) ^2 - 2\cdot s^2 \cdot 32 ^2 \\ &= \left( s^2 + 32 \right) ^2 - 64s^2 \\ &= \left( s^2 + 32 \right) ^2 -...
  41. Rapier

    Laplace Transforms with an Undamped Spring with Unit Impulse

    Ok, so I start out with the basics and find k. F=ma=kx (1.5 kg)(9.8 m/s^2) = k (4.9 m) k = 3 N/m I also know from the problem that c=0 (no damping) and x(0) = 2m and x'(0) = 0. ƩF = ma ma = -cx' - kx, a = x'' mx'' + cx' + kx = 0 and since c=0 mx'' + kx = 0 Nothing out of the...
  42. P

    How Do Laplace Transforms Help Understand Frequency and Amplitude in Signals?

    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...
  43. A

    Laplace transforms for the transient analysis of networks

    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...
  44. A

    Laplace transforms for transient analysis

    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...
  45. T

    Laplace Transform - Learn & Calculate Transforms Easily

    Please ignore this thread.
  46. T

    Trying to find inverse laplace transforms for some problems

    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...
  47. C

    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. J

    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. E

    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. Chris L T521

    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.
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