Recent questions in Laplace transform

Differential EquationsAnswered question

Eliza Shields 2023-04-01

The Laplace transform of $u(t-2)$ is

(a)$\frac{1}{s}+2$

(b)$\frac{1}{s}-2$

(c)$e}^{2}\frac{s}{s}\left(d\right)\frac{{e}^{-2s}}{s$ ??

(a)

(b)

(c)

Differential EquationsAnswered question

Gianna Johnson 2023-04-01

The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True or False

Differential EquationsAnswered question

inframundosa921 2023-03-21

The Laplace transform of $t{e}^{t}$ is A. $\frac{s}{(s+1{)}^{2}}$ B. $\frac{1}{(s-1{)}^{2}}$ C. $\frac{s}{(s+1{)}^{2}}$ D. $\frac{s}{(s-1)}$

Differential EquationsAnswered question

enrosca0fx 2023-01-03

What is the Laplace transform of $t\mathrm{cos}t$ into the s domain?

Differential EquationsAnswered question

klupko5HR 2022-11-25

Find the inverse Laplace transform of $\frac{{s}^{2}-4s-4}{{s}^{4}+8{s}^{2}+16}$

Differential EquationsAnswered question

phumzaRdY 2022-11-25

How do i find the lapalace transorm of this intergral using the convolution theorem? ${\int}_{0}^{t}{e}^{-x}\mathrm{cos}x\phantom{\rule{thinmathspace}{0ex}}dx$

Differential EquationsAnswered question

unecewelpGGi 2022-11-25

What's the correct way to go about computing the Inverse Laplace transform of this?

$\frac{-2s+1}{({s}^{2}+2s+5)}$

I Completed the square on the bottom but what do you do now?

$\frac{-2s+1}{(s+1{)}^{2}+4}$

$\frac{-2s+1}{({s}^{2}+2s+5)}$

I Completed the square on the bottom but what do you do now?

$\frac{-2s+1}{(s+1{)}^{2}+4}$

Differential EquationsAnswered question

hemotropS7A 2022-11-25

How to find inverse Laplace transform of the following function?

$X(s)=\frac{s}{{s}^{4}+1}$

I tried to use the definition: $f(t)={\mathcal{L}}^{-1}\{F(s)\}=\frac{1}{2\pi i}\underset{T\to \mathrm{\infty}}{lim}{\int}_{\gamma -iT}^{\gamma +iT}{e}^{st}F(s)\phantom{\rule{thinmathspace}{0ex}}ds$or the partial fraction expansion but I have not achieved results.

$X(s)=\frac{s}{{s}^{4}+1}$

I tried to use the definition: $f(t)={\mathcal{L}}^{-1}\{F(s)\}=\frac{1}{2\pi i}\underset{T\to \mathrm{\infty}}{lim}{\int}_{\gamma -iT}^{\gamma +iT}{e}^{st}F(s)\phantom{\rule{thinmathspace}{0ex}}ds$or the partial fraction expansion but I have not achieved results.

Differential EquationsAnswered question

Alberanteb4T 2022-11-24

inverse laplace transform - with symbolic variables:

$F(s)=\frac{2{s}^{2}+(a-6b)s+{a}^{2}-4ab}{({s}^{2}-{a}^{2})(s-2b)}$

My steps:

$F(s)=\frac{2{s}^{2}+(a-6b)s+{a}^{2}-4ab}{(s+a)(s-a)(s-2b)}$

$=\frac{A}{s+a}+\frac{B}{s-a}+\frac{C}{s-2b}+K$

$K=0$

$A=F(s)\ast (s+a)$

$F(s)=\frac{2{s}^{2}+(a-6b)s+{a}^{2}-4ab}{({s}^{2}-{a}^{2})(s-2b)}$

My steps:

$F(s)=\frac{2{s}^{2}+(a-6b)s+{a}^{2}-4ab}{(s+a)(s-a)(s-2b)}$

$=\frac{A}{s+a}+\frac{B}{s-a}+\frac{C}{s-2b}+K$

$K=0$

$A=F(s)\ast (s+a)$

Differential EquationsAnswered question

ingerentayQL 2022-11-24

Laplace transform of $(3e{)}^{t}{\mathrm{sin}}^{2}t$

Differential EquationsAnswered question

hemotropS7A 2022-11-24

How to find the Direct Discrete Laplace Transform of $(}\genfrac{}{}{0ex}{}{2n}{n}{\textstyle )$

Differential EquationsOpen question

Diana karen Sánchez González2022-11-22

Trans

y''+ 5y'+ 6y = 0, y(0) = 1, y'(0) = 1

Differential EquationsAnswered question

Leonard Dyer 2022-11-22

Find Inverse Laplace Transform of $\frac{1}{({s}^{2}+1)({s}^{2}-2s+7)}.$

Differential EquationsAnswered question

AimettiA8J 2022-11-22

How to find the Laplace transform of $|\mathrm{sin}(t)|$?

Differential EquationsAnswered question

undergoe8m 2022-11-21

Show ${\int}_{s}^{\mathrm{\infty}}f(x)dx=\mathcal{L}\{\frac{F(t)}{t}\}$ given $f(x)={\int}_{0}^{\mathrm{\infty}}{e}^{-xt}F(t)dt$

Differential EquationsAnswered question

Celeste Barajas 2022-11-21

This integral sounds quite complex and I could not find an approximate equivalent.

${\int}_{0}^{+\mathrm{\infty}}x\mathrm{log}(1+{x}^{2})\phantom{\rule{thinmathspace}{0ex}}{e}^{-Bx}\phantom{\rule{thinmathspace}{0ex}}dx$

${\int}_{0}^{+\mathrm{\infty}}x\mathrm{log}(1+{x}^{2})\phantom{\rule{thinmathspace}{0ex}}{e}^{-Bx}\phantom{\rule{thinmathspace}{0ex}}dx$

Differential EquationsAnswered question

figoveck38 2022-11-21

Solve ${y}^{\prime}(t)=\mathrm{sin}(t)+{\int}_{0}^{t}y(x)\mathrm{cos}(t-x)dx$ by Laplace transform

My try:

I applied Laplace transform on both sides of the equation.

$sL\{y(t)\}=\frac{1}{{s}^{2}+1}+L\{cos(t)\ast y(t)\}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}sL\{y(t)\}=\frac{1}{{s}^{2}+1}+L\{cos(t)\}\times L\{y(t)\}$

Now, I'm stuck on applying the inverse Laplace transform on (*) to find $y(t)$

My try:

I applied Laplace transform on both sides of the equation.

$sL\{y(t)\}=\frac{1}{{s}^{2}+1}+L\{cos(t)\ast y(t)\}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}sL\{y(t)\}=\frac{1}{{s}^{2}+1}+L\{cos(t)\}\times L\{y(t)\}$

Now, I'm stuck on applying the inverse Laplace transform on (*) to find $y(t)$

Differential EquationsAnswered question

Noe Cowan 2022-11-20

Laplace transform of ${t}^{2}{e}^{at}$??

try to prove that

$\mathcal{L}\{{t}^{2}{e}^{at}\}=\frac{2}{(s-a{)}^{3}}.$

I've gotten to the last integration by parts where

$\underset{n\to \mathrm{\infty}}{lim}{\int}_{0}^{n}\frac{1}{(a-s{)}^{2}2{e}^{(a-s)t}}dt={\underset{n\to \mathrm{\infty}}{lim}\frac{2}{(a-s{)}^{3}}{e}^{(a-s)t}|}_{0}^{n}.$

Now what do I do?

try to prove that

$\mathcal{L}\{{t}^{2}{e}^{at}\}=\frac{2}{(s-a{)}^{3}}.$

I've gotten to the last integration by parts where

$\underset{n\to \mathrm{\infty}}{lim}{\int}_{0}^{n}\frac{1}{(a-s{)}^{2}2{e}^{(a-s)t}}dt={\underset{n\to \mathrm{\infty}}{lim}\frac{2}{(a-s{)}^{3}}{e}^{(a-s)t}|}_{0}^{n}.$

Now what do I do?

Differential EquationsAnswered question

Kenna Stanton 2022-11-20

$\underset{s\to {0}^{+}}{lim}{\int}_{0}^{\mathrm{\infty}}a(t){e}^{-st}dt$

${\int}_{0}^{\mathrm{\infty}}a(t){e}^{-st}dt=f(s)$

What is the meaning of the limit of this integral as $s\to {0}^{+}.$

${\int}_{0}^{\mathrm{\infty}}a(t){e}^{-st}dt=f(s)$

What is the meaning of the limit of this integral as $s\to {0}^{+}.$

Differential EquationsAnswered question

Widersinnby7 2022-11-20

Solving the IVP

$t{y}^{\u2033}-t{y}^{\prime}+y=1$

where $y(0)=0$ and ${y}^{\prime}(0)=2$

$t{y}^{\u2033}-t{y}^{\prime}+y=1$

where $y(0)=0$ and ${y}^{\prime}(0)=2$

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.