Recent questions in Laplace transform

Differential EquationsAnswered question

trumansoftjf0 2022-11-11

Find the inverse Laplace transform of the following:

$F(s)=\frac{2s+1}{{s}^{2}-2s+2}.$

The answer is $f(t)=2{e}^{t}\mathrm{cos}t+3{e}^{t}\mathrm{sin}t$

Obviously once you have the decomposed fraction the remainder of the problem is simple but I can't seem to get to that point. Could someone please lay out the steps to decompose F(s)?

$F(s)=\frac{2s+1}{{s}^{2}-2s+2}.$

The answer is $f(t)=2{e}^{t}\mathrm{cos}t+3{e}^{t}\mathrm{sin}t$

Obviously once you have the decomposed fraction the remainder of the problem is simple but I can't seem to get to that point. Could someone please lay out the steps to decompose F(s)?

Differential EquationsAnswered question

perlejatyh8 2022-11-11

Evaluating the inverse laplace transform of $X(s)=\frac{2\cdot {a}^{4}\cdot s}{{s}^{4}+4\cdot {a}^{4}}$

Differential EquationsAnswered question

nyle2k8431 2022-11-11

Calculate $\underset{s\to 0}{lim}\text{}{\int}_{0}^{+\mathrm{\infty}}\frac{{e}^{-st}}{1+{t}^{2}}\text{}dt$

$\underset{s\to 0}{lim}\text{}{\int}_{0}^{+\mathrm{\infty}}\frac{{e}^{-st}}{1+{t}^{2}}\text{}dt$

$\mathcal{L}\{\mathrm{arctan}(t)\}=\frac{F(s)}{s}$

Final value theorem:

$\underset{s\to 0}{lim}\text{}sF(s)=\underset{t\to +\mathrm{\infty}}{lim}f(t)$

$\underset{t\to +\mathrm{\infty}}{lim}\text{}\mathrm{arctan}(t)=\frac{\pi}{2}$

$\underset{s\to 0}{lim}\text{}\text{}s\text{}\frac{1}{s}F(s)=\frac{\pi}{2}$

Is it correct?

$\underset{s\to 0}{lim}\text{}{\int}_{0}^{+\mathrm{\infty}}\frac{{e}^{-st}}{1+{t}^{2}}\text{}dt$

$\mathcal{L}\{\mathrm{arctan}(t)\}=\frac{F(s)}{s}$

Final value theorem:

$\underset{s\to 0}{lim}\text{}sF(s)=\underset{t\to +\mathrm{\infty}}{lim}f(t)$

$\underset{t\to +\mathrm{\infty}}{lim}\text{}\mathrm{arctan}(t)=\frac{\pi}{2}$

$\underset{s\to 0}{lim}\text{}\text{}s\text{}\frac{1}{s}F(s)=\frac{\pi}{2}$

Is it correct?

Differential EquationsAnswered question

Yaretzi Mcconnell 2022-11-11

How to solve the following integral:

$I={\int}_{0}^{\mathrm{\infty}}\mathrm{cos}(t\mathrm{l}\mathrm{o}\mathrm{g}(x))\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{-ax}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x,$

where t and a are real.

$I={\int}_{0}^{\mathrm{\infty}}\mathrm{cos}(t\mathrm{l}\mathrm{o}\mathrm{g}(x))\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{-ax}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x,$

where t and a are real.

Differential EquationsAnswered question

Humberto Campbell 2022-11-11

If $y(t)={\int}_{0}^{t}f(t)dt$ & the Laplace transform of f(t) is $\mathcal{L}\{f(t)\}={\int}_{0}^{\mathrm{\infty}}{e}^{-st}f(t)dt$,then prove that $\mathcal{L}\{y(t)\}=(1/s)\mathcal{L}\{f(t)\}$

Differential EquationsAnswered question

Kailyn Hamilton 2022-11-11

Show using Laplace Transform that:

${\int}_{0}^{\mathrm{\infty}}\text{}\mathrm{cos}({x}^{2})\text{}dx=\frac{1}{2}\sqrt{\frac{\pi}{2}}$

The right hand side seems like it has something to do with the gamma function. Help with it please!

${\int}_{0}^{\mathrm{\infty}}\text{}\mathrm{cos}({x}^{2})\text{}dx=\frac{1}{2}\sqrt{\frac{\pi}{2}}$

The right hand side seems like it has something to do with the gamma function. Help with it please!

Differential EquationsAnswered question

Josie Kennedy 2022-11-10

Solve by Laplace Transforms.

How to find this ${\mathcal{L}}^{-1}$ $(\frac{\frac{5s}{4}+\frac{13}{4}}{{s}^{2}+5s+8})$

How to find this ${\mathcal{L}}^{-1}$ $(\frac{\frac{5s}{4}+\frac{13}{4}}{{s}^{2}+5s+8})$

Differential EquationsAnswered question

Nola Aguilar 2022-11-10

Show that

${\mathcal{L}}^{-1}\left[\frac{f(s)}{{s}^{2}}\right]={\int}_{0}^{t}{\int}_{0}^{x}F(x)dxdy.$

I tried using the formula

${\mathcal{L}}^{-1}\left[\frac{f(s)}{s}\right]={\int}_{0}^{t}F(x)dx.$

${\mathcal{L}}^{-1}\left[\frac{f(s)}{{s}^{2}}\right]={\int}_{0}^{t}{\int}_{0}^{x}F(x)dxdy.$

I tried using the formula

${\mathcal{L}}^{-1}\left[\frac{f(s)}{s}\right]={\int}_{0}^{t}F(x)dx.$

Differential EquationsAnswered question

unabuenanuevasld 2022-11-10

Why is the inverse Laplace Transform of

$\frac{\mathrm{sinh}(x\sqrt{s})}{s\cdot \mathrm{sinh}\sqrt{s}}$

equal to

$x+\frac{2}{\pi}\sum _{n=1}^{\mathrm{\infty}}\frac{(-1{)}^{n}}{n}{e}^{-(n\pi {)}^{2}t}\mathrm{sin}n\pi x?$

$\frac{\mathrm{sinh}(x\sqrt{s})}{s\cdot \mathrm{sinh}\sqrt{s}}$

equal to

$x+\frac{2}{\pi}\sum _{n=1}^{\mathrm{\infty}}\frac{(-1{)}^{n}}{n}{e}^{-(n\pi {)}^{2}t}\mathrm{sin}n\pi x?$

Differential EquationsAnswered question

Layton Park 2022-11-10

Use Laplace Transform to solve the following IVP:

${y}^{\u2033}+2{y}^{\prime}+5y={e}^{-t}\mathrm{sin}(2t)$ where $y(0)=2,{y}^{\prime}(0)=-1$

${y}^{\u2033}+2{y}^{\prime}+5y={e}^{-t}\mathrm{sin}(2t)$ where $y(0)=2,{y}^{\prime}(0)=-1$

Differential EquationsAnswered question

klesstilne1 2022-11-10

$$F(S)=\frac{-S+11}{{S}^{2}-2S-3}$$

How do I find f(t)? What is a good strategy for attacking these types of problems?

How do I find f(t)? What is a good strategy for attacking these types of problems?

Differential EquationsAnswered question

Josie Kennedy 2022-11-09

How can we find the inverse Laplace transform of the function

$$H(s)=\frac{8}{{s}^{4}+4}?$$

$$H(s)=\frac{8}{{s}^{4}+4}?$$

Differential EquationsAnswered question

Jenny Roberson 2022-11-09

How can I find the Inverse Laplace Transform of : ${\left(\frac{1-{s}^{1/2}}{{s}^{2}}\right)}^{2}$

Differential EquationsAnswered question

linnibell17591 2022-11-09

How to solve this Laplace transform? $f(t)={e}^{-2t}{\mathrm{cos}}^{2}3t-3{t}^{2}{e}^{3t}$

The answer is

$\frac{1}{2(s+2)}+\frac{1}{2}\frac{s+2}{{s}^{2}+4s+40}-\frac{6}{(s-3{)}^{3}}.$

How can this be solved?

The answer is

$\frac{1}{2(s+2)}+\frac{1}{2}\frac{s+2}{{s}^{2}+4s+40}-\frac{6}{(s-3{)}^{3}}.$

How can this be solved?

Differential EquationsAnswered question

fabler107 2022-11-08

Evaluate:

$${\int}_{0}^{\mathrm{\infty}}\frac{(1-{\text{e}}^{-px})(1-{\text{e}}^{-qx})(1-{\text{e}}^{-rx})}{{\text{e}}^{x}}\text{d}x,\text{}\text{}\text{}p0,\text{}q0,\text{}r0$$

I try it with laplace transfrom, but I cant find a result

$${\int}_{0}^{\mathrm{\infty}}\frac{(1-{\text{e}}^{-px})(1-{\text{e}}^{-qx})(1-{\text{e}}^{-rx})}{{\text{e}}^{x}}\text{d}x,\text{}\text{}\text{}p0,\text{}q0,\text{}r0$$

I try it with laplace transfrom, but I cant find a result

Differential EquationsAnswered question

Rosemary Chase 2022-11-08

Finding the inverse Laplace of this function $\frac{1}{{({s}^{2}+1)}^{2}}$

Differential EquationsAnswered question

Humberto Campbell 2022-11-08

Find the Laplace transform of $g(t)=1+co{s}^{2}(2t)$ by direct integration

I'm having trouble finding out how to directly integrate the function f(t) because of the ${\mathrm{cos}}^{2}(2t)$ term. I understand that ${\mathrm{cos}}^{2}(2t)=\frac{1}{2}+\frac{1}{2}\mathrm{cos}(4t)$ but I don't understand how this simplifies the problem so that

${\int}_{0}^{\mathrm{\infty}}{e}^{-st}+{\int}_{0}^{\mathrm{\infty}}{e}^{-st}(\frac{1}{2}+\frac{1}{2}\mathrm{cos}(4t))$

Is an easier integral to solve.

I'm having trouble finding out how to directly integrate the function f(t) because of the ${\mathrm{cos}}^{2}(2t)$ term. I understand that ${\mathrm{cos}}^{2}(2t)=\frac{1}{2}+\frac{1}{2}\mathrm{cos}(4t)$ but I don't understand how this simplifies the problem so that

${\int}_{0}^{\mathrm{\infty}}{e}^{-st}+{\int}_{0}^{\mathrm{\infty}}{e}^{-st}(\frac{1}{2}+\frac{1}{2}\mathrm{cos}(4t))$

Is an easier integral to solve.

Differential EquationsAnswered question

Kayley Dickson 2022-11-08

What is the easiest way to find the inverse Laplace of F(s)? $$F(s)=\frac{1}{(s-1{)}^{2}(1-1/{s}^{2})}$$

Differential EquationsAnswered question

Kenna Stanton 2022-11-08

How do I prove that the Bessel function ${J}_{0}=\frac{1}{\pi}{\int}_{0}^{\pi}\mathrm{cos}(x\mathrm{cos}(t))dt$ using Laplace transforms?

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.