Explore Laplace Transform Equations

Use properties of the Laplace transform to answer the following
(a) If $f (t) = (t + 5)^{2} + t^{2} e^{5 t}$ , find the Laplace transform, $L [f (t)] = F (s)$ .
(b) If $f (t) = 2 e^{- t} \cos (3 t + \frac{π}{4})$ , find the Laplace transform, $L [f (t)] = F (s)$ . HINT:
$\cos (α + β) = \cos (α) \cos (β) - \sin (α) \sin (β)$
(c) If $F (s) = \frac{7 s^{2} - 37 s + 64}{s (s^{2} - 8 s + 16)}$ find the inverse Laplace transform, $L^{- 1} | F (s) | = f (t)$
(d) If $F (s) = e^{- 7 s} (\frac{1}{s} + \frac{s}{s^{2} + 1})$ , find the inverse Laplace transform, $L^{- 1} [F (s)] = f (t)$

FizeauV 2021-02-24

What would be the laplace transform of a function of f or $L {f (t)}$ ?

ediculeN 2021-02-24

For the function
$f (t) = e^{t}$
$g (t) = e^{- 2 t}$
$0 \leq t < \infty$
compute in two different ways:
a) By directly evaluating the integral in the defination of $f \cdot g$
b) By computing $L^{- 1} {F (s) G (s)} where F (s) = L {f (t)} and G (s) = L {g (t)}$

banganX 2021-02-24

determine the inverse Laplace transform of F.
$F (s) = \frac{e^{- 2 s}}{(s - 3)^{3}}$

Kye 2021-02-21

Find the Laplace transforms of the following time functions.
Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables.
a) $f (t) = 1 + 2 t$ b) $f (t) = \sin ω t Hint: Use Euler’s relationship, \sin ω t = \frac{e^{(} j ω t) - e^{(} - j ω t)}{2 j}$
c) $f (t) = \sin (2 t) + 2 \cos (2 t) + e^{- t} \sin (2 t)$

ankarskogC 2021-02-21

Solve the initial value problem below using the method of Laplace transforms.
$y " - 16 y = 32 t - 8 e^{- 4 t}$
$y (0) = 0$
$y^{'} (0) = 15$

opatovaL 2021-02-21

Find the Laplace transforms of the given functions.
$g (t) = 4 \cos (4 t) - 9 \sin (4 t) + 2 \cos (10 t)$

Chesley 2021-02-21

Use the Laplace transform to solve the heat equation
$u_{t} = u_{x x} 0 < x < 1 and t > 0$
$u (x, 0) = \sin (π x)$
$u (0, t) = u (1, t) = 0$

Joni Kenny 2021-02-20

Find the inverse Laplace transform of (any two)
i) $\frac{(s^{2} + 3)}{s (s^{2} + 9)}$
ii) $\log (\frac{(s + 1)}{(s - 1)})$

glasskerfu 2021-02-19

Solve the differential equation using Laplace transform of
$y^{″} - 3 y^{'} + 2 y = e^{3 t}$
when y(0)=0 and y'(0)=0

Chardonnay Felix 2021-02-19

Find the inverse Laplace transform of
$\frac{e^{- s}}{s (s + 1)}$

tricotasu 2021-02-19

Find the inverse laplace trans.
$F (s) = \frac{10}{s (s^{2} + 9)}$

Clifland 2021-02-19

Use Laplace transform to solve the following initial-value problem
$y " + 2 y^{'} + y = 0$
$y (0) = 1, y^{'} (0) = 1$
a) $e^{- t} + t e^{- t}$
b) $e^{t} + 2 t e^{t}$
c) $e^{- t} + 2 t e^{t}$
d) $e^{- t} + 2 t e^{- t}$
e) $2 e^{- t} + 2 t e^{- t}$
f) Non of the above

floymdiT 2021-02-16

find the inverse Laplace transform of the given function.
$F (s) = \frac{2 s - 3}{s^{2} - 4}$

BolkowN 2021-02-16

Find Laplace transform
$L {(t - 3)^{2} u (t - 3)}$

Daniaal Sanchez 2021-02-16

Solution of I.V.P for harmonic oscillator with driving force is given by Inverse Laplace transform
$y " + ω^{2} y = \sin γ t, y (0) = 0, y^{'} (0) = 0$
1) $y (t) = L^{- 1} (\frac{γ}{(s^{2} + ω^{2})^{2}})$
2) $y (t) = L^{- 1} (\frac{γ}{s^{2} + ω^{2}})$
3) $y (t) = L^{- 1} (\frac{γ}{(s^{2} + γ^{2})^{2}})$
4) $y (t) = L^{- 1} (\frac{γ}{(s^{2} + γ^{2}) (s^{2} + ω^{2})})$

remolatg 2021-02-16

$Find the Laplace transform F (s) = L {f (t)} of the function f (t) = 6 + \sin (3 t) defined on the interval t \geq 0$

he298c 2021-02-15

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
$y " - 4 y' = - 4 t e^{2 t}, y_{0} = 0, y'_{0} = 1$
$y " + 2 y' + 10 y = - 6 e^{- t} \sin 3 t, y_{0} = 0, y'_{0} = 1$

facas9 2021-02-14

Use Laplace transforms to solve the following initial value problem
$y " - y^{'} - 6 y = 0$
$y (0) = 1$
$y^{'} (0) = - 1$

necessaryh 2021-02-14

$y " - 8 y^{'} + 41 y = 0$
$y (0) = 0$
$y^{'} (0) = 5$
Now, using Y for the Laplace transform of y(t), i.e., $Y = L {y (t)}$ find the equation that results from the differential equation's Laplace transformation.

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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