Explore Laplace Transform Equations

Need to find the integral of the following function using Laplace Transform
$\int_{0}^{\infty} \frac{\cos (6 t) - \cos (4 t)}{t} d t$

steveo963200054 2022-09-14

How to prove $P {min (X_{1}, X_{2}, \dots, X_{n}) = X_{i}} = \frac{λ_{i}}{λ_{1} + \dots + λ_{n}}$ , when $X_{i}$ is exponentially distributed

lemondedeninaug 2022-09-14

$L ((t + 5) U (t - 1)) = U (t - a) f (t - a) = e^{- a s} F (s)$
a= 1
I am having trouble dealing with $f (t - a)$ and finding $F (s)$
$f (t - 1) = e^{t - 3} = e^{(t - 1) - 3 + 1} = e^{(t - 1) - 2}$
$f (t) = e^{t - 2}$
How do I Laplace transform this ?

Gaintentavyw4 2022-09-14

There are given equations,
$x = x^{'} \cos θ - y^{'} \sin θ y = x^{'} \sin θ + y^{'} \cos θ$
$\frac{\partial f}{\partial x^{'}} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial x^{'}} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial x^{'}}$
I don't understand how it is equal to
$\frac{\partial f}{\partial x^{'}} = \frac{\partial f}{\partial x} \cos θ + \frac{\partial f}{\partial y} \sin θ$
Can you explain?

Nyasia Flowers 2022-09-14

Why following equality is satisfied
$L [\exp (\int_{0}^{t} d s \frac{\dot{x} (s)}{x (s)})] = \frac{1}{u - \frac{L [\dot{x}] (u)}{L [x] (u)}},$
where $L [\cdot]$ denotes Laplace transform. Is this equation fulfilled for all x(t), and how one can derive it?

Dulce Cantrell 2022-09-14

How the answer was computed for the following infinite series:
$\sum_{x = 1}^{\infty} e^{- s x} p q^{x - 1} = \frac{p e^{- s}}{1 - q e^{- s}}$

Spactapsula2l 2022-09-14

Find the inverse Laplace transform of $F (s) = \frac{4 s}{(s^{2} + 5)^{2} - 4 s^{2}}$ , but I'm having trouble simplifying this fraction so I'm able to take the Laplace transform.
I simplified the denominator and got $s^{4} + 6 s^{2} + 25$ , but I'm not sure how that helps me.
I also tried looking at $4 s^{2}$ and $(2 s)^{2}$ , but still am stuck.

ridge041h 2022-09-14

How did the answer found the antiderivative of $\frac{a}{a^{2} + \cos^{2} x}$ to be $\frac{1}{\sqrt{1 + a^{2}}} \tan^{- 1} (\frac{a \tan x}{\sqrt{1 + a^{2}}})$ ?
I have not done Laplce transform previously and so is this purely by Laplace transform? I could verify that this is indeed the anti-derivative but could we get around Laplace transform to find the antiderivative straightaway?

Jonah Jacobson 2022-09-14

Solve the differential equation $y^{″} + 2 y^{'} + 5 y = \sin 2 t$ using Laplace transform when y=0 and y′=0 when t=0
I've done the transform part and got a function -
$y (S) = \frac{2}{(s^{2} + 4) (s^{2} + 2 s + 5)} = \frac{A}{s^{2} + 4} + \frac{B s + C}{s^{2} + 2 s + 5}$

steveo963200054 2022-09-13

Using Laplace Transforms, find the solution to $y^{'} (t) + y (t - 1) = t^{2}$ , with y(t)=0 for $t \leq 0$

pobi1k 2022-09-13

Identify the function f(t) that has the following Laplace transform,
$\tilde{f} (s) = \int_{0}^{\infty} f (t) e^{- s t} d t = {(\frac{1 + α s}{1 + α (s - s_{0})})}^{p}$
where $α, s_{0}, p$ are positive parameters. Any suggestions about how to approach this problem?

Nyasia Flowers 2022-09-13

Solve the following integral:
$\int_{0}^{\infty} \frac{1}{x} e^{- \frac{x}{a}} (e^{- e^{- (\frac{x - c}{b})}}) d x$
I wanted to compute it by using Laplace Transform, but $\frac{1}{x}$ part made me confused and I could not reach to a solution.

Kaleigh Ayers 2022-09-13

Find $L^{- 1} (F (s))$ , if given $F (s) = \frac{2}{s (s^{2} + 4)}$
I have tried as below.
To find inverse of Laplace transform, I want to make partial fraction as below.
$\begin{aligned} \frac{2}{s (s^{2} + 4)} = \frac{A}{s} + \frac{B s + C}{s^{2} + 4} = \frac{(A + B) s^{2} + C s + 4 A}{s (s^{2} + 4)} . \end{aligned}$
After that, we have system of linear equation
$\begin{aligned} A + B & = 0 \\ C & = 0 \\ 4 A & = 2. \end{aligned}$
Thus we have A=2, B=−2, and C=0. Now, substituting A,B,C and we have
$\begin{aligned} \frac{2}{s (s^{2} + 4)} = \frac{2}{s} + \frac{- 2 s}{s^{2} + 4} . \end{aligned}$
But the fact is
$\begin{aligned} \frac{2}{s (s^{2} + 4)} \neq \frac{2}{s} + \frac{- 2 s}{s^{2} + 4} = \frac{8}{s^{2} + 4} . \end{aligned}$
I'm stuck here. I can't make a partial fraction for F(s) and I can't find inverse of Laplace transform for F(s).
Anyone can give me hint to give me hint for this problem?

katdoringlo 2022-09-13

Need to convert the following equation from Laplace domain to continuous time domain:
$F (s) = \frac{- 2 m k v R}{2 m R s^{2} + m k s + 2 k R}$
m, k, v, R are all constants.

skystasvs 2022-09-13

What is Laplace transform of $\frac{e^{- t}}{t} \sin 3 t \sin 2 t$

moidu13x8 2022-09-12

Find the following inverse Laplace transform.
$L^{- 1} [\ln (1 + \frac{4 a}{s^{2}})]$
I don't know because of 4a.

Paul Reilly 2022-09-12

How to calculate the following Laplace transform: $L [\frac{1 - J_{0} (t)}{t}]$ ? where $J_{0} (t)$ is the zeroth Bessel function.

profesorluissp 2022-09-12

Need to evaluate the Laplace integral transform (definition below) of $\sqrt{t}$ using complex analysis. How is that possible?
$\hat{f} (s) = \int_{0}^{\infty} \sqrt{t} e^{- s t} d t$

Lina Neal 2022-09-12

I know that
$L (t^{2}) = \frac{2}{s^{3}}$
but why is is that, if we use the convolution theorem for $1 * t^{2}$ , we get
$L (1 * t^{2}) = \frac{1}{s} * \frac{2}{s^{3}}$
Isn't $1 * t^{2}$ equal to $t^{2}$ ?

sincsenekdq 2022-09-12

$I = \int_{0}^{\infty} \frac{\sin (k x)}{x (x^{2} + 1)} d x$
Where $k \in R^{+}$
What methods can be employed to solve this integral?

If you came across the necessity of Laplace transform, it is most likely that you are coming from a mechanical engineering or electrical background. The concept is used to solve differential equations, which is why it is vital to consider Laplace transform examples as you are looking through the questions and connect the dots as the equations are being approached. Remember to look through our list of answers as these will help you to address various Laplace transform problems and find solutions to complex Laplace transform questions as you are dealing with your Laplace transform equation homework.

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