Explore Laplace Transform Equations

If F(s) is the Laplace transform of $f (t) = t^{- 3 / 2} e^{- 1 / t}$ , show that $F^{'} (s) = - s^{- 1 / 2} F (s)$

tonan6e 2022-09-03

How to find the Laplace Transform of $t^{2} s i n (t)$
Using the rule:
$L (t^{n} f (t)) = (- 1)^{n} \frac{d^{n}}{d s^{n}} F (s)$
where in this case
$f (t) = \sin (t), L (\sin (t)) = F (s) = \frac{1}{s^{2} + 1}, n = 2.$
Find the 2nd derivative of F(s):
$\frac{d^{2}}{d s^{2}} (\frac{1}{s^{2} + 1}) = \frac{6 s^{2} - 2}{(s^{2} + 1)^{3}}$
The transform:
$L (t^{2} s i n (t)) = (- 1)^{2} \frac{6 s^{2} - 2}{(s^{2} + 1)^{3}}$

clovnerie0q 2022-09-03

Using Laplace transform, solve the system:
$w^{'} + y = \sin (x)$
$y^{'} - z = e^{x}$
$z^{'} + w + y = 1$
where w(0)=0 and z(0)=y(0)=1.

Riya Andrews 2022-09-03

Using laplace trasnform find the convulation of (fg)(t).
$l e t f (t) = \sin (3 t)$ and $g (t) = e^{- 2 t}$
Using laplace trasnform find the convulation of (fg)(t).
convulation theorem: $h (t) = (f * g) (t) = \int_{0}^{t} f (u) g (t - u) d u$
$L (s i n (3 t) = \frac{3}{s^{2} + 9} L (e^{- 2 t}) = \frac{1}{s + 2}$
then how we processed for this problem

Kaleigh Ayers 2022-09-03

Use the Laplace transform table and the linearity of the Laplace transform to determine the following:
$L {t^{4} - t^{2} - t + \sin \sqrt{2 t}}$

Scamuzzig2 2022-09-01

I worked out this far and I am having trouble with the integral
$F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t$
$f (t) = e^{t + 7}$
$\int_{0}^{\infty} e^{- (s - 1) t} e^{7} d t$
we can say that $e^{7} = c_{1}$ and show
$- \frac{c_{1}}{s - 1} \int_{0}^{\infty} e^{- (s - 1) t} d t$
I would surmise that I can take the limit at this point. So
$- \frac{c_{1}}{s - 1} {lim}_{b}^{\infty} \int_{0}^{b} e^{- (s - 1) t} d t$
which is likely evaluates as
$- \frac{c_{1}}{s - 1} (0 - 1)$
yielding
$\frac{e^{7}}{s - 1}$
What is "good practice" that I have not shown?

emmostatwf 2022-09-01

I have an expression such as $\int_{0}^{x + l} y (z) g (x - z) d z$ and I want to evaluate its Laplace transform w.r.t x in terms of the Laplace transform of y(x). I know that I can substitute $t = x + l$ , and coerce it into the standard from to get the Laplace transform w.r.t t, but I need its transform w.r.t x.
Motivation:
I would like to solve an integral equation: $y (x) = f (x) + \int_{0}^{x + l} y (z) g (x - z) d z$
If the integral limit had been to x, we would have had $y (x) = f (x) + \int_{0}^{x} y (z) g (x - z) d z$ . This leads to $Y (s) = \frac{F (s)}{1 + K (s)}$

Jaxson Mack 2022-08-07

For the transformation $L : C^{1} [a, b] \to$ $C [a, b]$ definied by $L (f) = \frac{d f}{d x}$ (means, derivative of function $f$ w.r.t $x$ ),
a) Show that $L$ is linear.
b) Find $K e r (L)$ .
c) Find $n u l l i t y (L)$ .
d) Is $L$ one-to-one? Explain.
e) is $L$ invertible? Explain.

Makayla Eaton 2022-08-05

Use Laplace transform to solve the given initial-value problem.
$\frac{d y}{d t} - y = 1, y (0) = 0$

kislotd 2022-07-25

Find the Laplace Transform of $L {(\sin h t) / t}$

iarc6io 2022-07-17

Using Laplace transforms calculate the solution of the ordinary differential equation $\frac{d^{2} y}{d t^{2}} - 2 \frac{d y}{d t} + y = e^{- t}$ , subject to the initial conditions y(0)=0 and $\frac{d y}{d t} (0) = 1$ . Select the correct answer from the following list.
$a) y (t) = \frac{3 t e^{t}}{2} + \frac{e^{- t}}{4} - \frac{e^{t}}{4} b) y (t) = t e^{t} + \frac{t^{2} e^{t}}{2} c) y (t) = \frac{t e^{- t}}{2} - \frac{e^{- t}}{4} + \frac{e^{t}}{4} d) y (t) = t e^{- t} + \frac{t^{2} e^{- t}}{2} e) y (t) = \frac{t e^{- t}}{2} - \frac{e^{- t}}{2} + \frac{e^{t}}{2}$

Peter Githeri2022-06-15

ashish.singh2021 2022-05-04

y''+4y= f(t); y(0)= 1, y''(0)=0 , with
f(t) = { 0 for 0≤t<4 , 3 for t≥4}

Jazmine Sweeney 2022-04-30

$\int_{0}^{\infty} {\cos (x t) o v e r 1 + t^{2}} d t$
I'm supposed to solve this using Laplace Transformations.

y"+4y= $\int_{1}^{0} 0 \leq t \leq t \leq \infty$
y(0)=1
y'(0)=0

Magdalena Norton 2022-04-21

Inverse Laplace Transform without using equations
I'm stuck with a question about Inverse Laplace Transform here, but the use of inverse laplace transform equation is forbidden.
$Y (s) = \frac{e^{- π s}}{s [{(s + 1)}^{2} + 1]}$

Quinn Moses 2022-04-16

Laplace transform of $f (t^{2})$
Suppose we know the Laplace transform of a function f(t):
$F (s) = \int_{0}^{\infty} f (t) e^{- s t} d t$

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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