# laplace transform Definition and Topics - 41 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

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

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1. ### I Laplace Transform of cos(t)/t

So, I know the direct definition of the Laplace Transform: $$\mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$ So when I plug in: $$\frac{\cos(t)}{t}$$ I get a divergent integral. however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29 is supposed to be the...

15. ### A Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)

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...
16. ### Engineering Math: Laplace Transform

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!!
17. ### A Triple Product in Laplace Transform

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...
18. ### A Initial value ODE with shifting forcing function

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
19. ### The Dirac Delta Function

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...
20. ### 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...
21. ### I How to find Y(s)/X(s)

y(t) = u(t - a) . x(t) u(t) is a unit step function. I have to find Y(s)/X(s). Do I have to do convolution in frequency domain?
22. ### Dartmouth Extended Laplace Tables -- Not general enough? item26.a

Homework Statement [/B] http://www.dartmouth.edu/~sullivan/22files/New Laplace Transform Table.pdf (see item 26a) homogenous solution to underdamped in amplitude phase form: (see attached image) 2. Relevant info - non zero initial conditions: x(t=0) = xo AND dx/dt(t=0) = vo - unforced...
23. ### Help with Laplace Transform

Homework Statement Diagram for a vehicle suspension is given. Displacement of wheel is given by 'x' and and displacement of body is 'y'. Spring constant, k = (7*10^4) Nm Damping coefficient, c = (3*10^3) N/m/s mass,m = 250kg a) Make a Laplace Transform of system and utilize it to predict 'y'...
24. ### Explicit check for Laplace transform?

Homework Statement Solve the following initial value problem using Laplace transforms: y' + 4y = 3t3 e−4t ; y(0) = 0 . Useful information: Recall that the Laplace transform of y 0 is pY − y(0), where Y is the Laplace Transform of y. The Laplace transform of tk e−at is k!/(p + a)k+1 . Confirm...
25. ### How would I take the laplace transform of f(t)= te^tsin^2(t)?

How would I take the laplace transform of f(t)= te^tsin^2(t)?
26. ### Laplace tranform of unitstep (heaviside) function

Homework Statement [/B] I know for t[u(t)-u(t-2)], we can simplify that to tu(t)-((t-2)+2)u(t-2) which then gives us tu(t)-(t-2)u(t-2)-u(t-2). Now, the laplace transform seems trivial but I am having problems with this equation: sin(t)[u(t)-u(t-pi)] Homework...
27. ### Laplace transform of y''(t')

The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this \begin{equation} y''(t')+2y''(t)-2y(t)=0 \end{equation} And the initial conditions are \begin{equation} y(0)=y'(0)=0, y''(0)=0 \end{equation} The problem is with the first...
28. ### Is there a mistake in my calculation or in my reasoning?

Homework Statement y'' + 3y' + 2y = r(t), r(t) = u(t - 1) - u(t - 2), y(0) = y'(0) = 0. I need to solve this by convolution, which I know is commutative. The problem is that my calculation gives (f * g) =/= (g * f). Could someone please tell me where my mistake is? Homework Equations...
29. ### Keep getting the wrong answer in this lengthy Laplace problem

Homework Statement y'' + 4y = 8t^2 if 0 < t < 5, and 0 if t > 5; y(1) = 1 + cos(2), y'(1) = 4 - 2sin(2). Use the Laplace transform to find y. Homework Equations t-shift, s-shift, unit step function. The Attempt at a Solution I have been trying to solve it for hours, but keep getting the wrong...
30. ### 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 -...