# Laplace transform integral problem

• TyErd
In summary: I would get.In summary, the conversation is discussing a problem involving Laplace transforms and inverse Laplace transforms. The group is trying to determine the best approach to solving the problem and there is some discussion about the correct formula to use and the limits of integration. Ultimately, the conversation ends with the correct solution of f(t) = 6(\frac{4t}{9} - \frac{5e^{-9t}}{81}+\frac{5}{81}).
TyErd

## Homework Statement

Hey guys, I've attached the question that's troubling me. I've also attached the table and formulas of Laplace transform for you convenience.

attached

## The Attempt at a Solution

Right now, I am thinking the best way to do this problem is by working backwards. I'm using the convolution integral formula.
I've already established that G(s) is 1/(s+4) using inverse Laplace transforms but I am not sure if that will get me an answer for f(t) rather F(s).

thats my inverse laplace------->invL
and laplace is ------> L

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When you take the Laplace transform of both sides of the equation, what are you getting? Where are you getting stuck?

I think I'm stuck because I'm not sure how to approach a problem like this. anyway, Laplace transform of both sides just makes it L{f(t)}= F(s)G(s) right? no that's not right coz there the 6t-5

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edit:
I am incorrect. Sorry.

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so that means h(t)= 6t so H(t) = 6/s^2

therefore
F(s) = 6/s^2 + 5/(s+4) F(s),
rearrange and F(s) = 6(s+4) / s^2(s-1)
and inverse Laplace it and f(t) = 6*(-4t + 5 e^t - 5)

is that right, I am not sure if my calculations are correct.

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i just saw your last post...r u sure ur wrong coz it actually made sense to me...

TyErd said:
i just saw your last post...r u sure ur wrong coz it actually made sense to me...

The limits of integration in the problem are from 0 to t, which makes me feel uncomfortable about my answer. If it were convolution, the limits would be -infinity to infinity. If f(u) is nonzero only for u > 0, then the limits could be from 0 to infinity. I'm just unsure why the top limit is t.

in the formula table I've provided, for convolution the limits says 0 to t.

TyErd said:
in the formula table I've provided, for convolution the limits says 0 to t.

If that's the case, then what I said made sense. I suppose G(i) is also zero for i < 0. So G(t-u) is zero outside u > t.

TyErd said:
so that means h(t)= 6t so H(t) = 6/s^2

therefore
F(s) = 6/s^2 + 5/(s+4) F(s),
rearrange and F(s) = 6(s+4) / s^2(s-1)
and inverse Laplace it and f(t) = 6*(-4t + 5 e^t - 5)

is that right, I am not sure if my calculations are correct.

$$F(s) = \frac{6}{s^2} - 5 \frac{1}{s+4} F(s)$$
Note the minus sign. Your next step would have been right if it had been addition (assuming a/bc is atually a/b/c). It is actually:
$$F(s) = 6\frac{s+4}{s^2(s+9)}$$

Your final answer also WOULD have been right without the sign error. Give it a try again with the proper minus sign this time.

ah i see so then the value of f(t) is 6 ((4t)/9 - 5e^(-9 t))/81 + 5/81 correct?

or (8*t)/3 - (10*exp(-9*t)/27) + 10/27

TyErd said:
ah i see so then the value of f(t) is 6 ((4t)/9 - 5e^(-9 t))/81 + 5/81 correct?

This has a lot of right terms in it, but it looks like you messed up on your parentheses.

$$6(\frac{4t}{9} - \frac{5e^{-9t}}{81}+\frac{5}{81})$$

## What is a Laplace transform integral problem?

A Laplace transform integral problem involves finding the Laplace transform of a given function, which is a mathematical operation that converts a function from the time domain to the frequency domain. This is useful in solving differential equations and analyzing systems in engineering and physics.

## What is the formula for the Laplace transform integral?

The formula for the Laplace transform integral is:
L[f(t)] = ∫0 e-st f(t) dt
where L[f(t)] represents the Laplace transform of the function f(t), s represents the complex frequency variable, and the integral is taken from 0 to infinity.

## What are the steps for solving a Laplace transform integral problem?

The steps for solving a Laplace transform integral problem are:
1. Determine the function f(t) that needs to be transformed.
2. Apply any necessary algebraic manipulations to the function to make it fit the form of the Laplace transform.
3. Use the Laplace transform table or properties to find the transform of the function.
4. Take the inverse Laplace transform, if necessary, to find the solution in the time domain.

## What are some common applications of Laplace transform integral problems?

Laplace transform integral problems are commonly used in engineering and physics to analyze systems and solve differential equations. They are also useful in signal processing, control systems, and circuit analysis.

## What are some tips for solving Laplace transform integral problems?

Some tips for solving Laplace transform integral problems include:
- Familiarize yourself with the Laplace transform table and properties.
- Make sure to properly manipulate the function to fit the form of the Laplace transform.
- Check your answer by taking the inverse Laplace transform and comparing it to the original function.
- Practice and review regularly to improve your understanding and speed in solving these problems.

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