Recent questions in Laplace transform

Differential EquationsAnswered question

Taylor Barron 2022-11-08

Solve ${L}^{-1}[\frac{1}{p({p}^{2}+4{)}^{2}}]$

Differential EquationsAnswered question

Kayden Mills 2022-11-07

Try to calculate the following inverse Laplace transform. $${\mathcal{L}}^{-1}\left\{\frac{1-1{e}^{-2s}}{s(s+2)}\right\}$$

Differential EquationsAnswered question

MMDCCC50m 2022-11-07

Find differential equation via Laplace transform $$x{y}^{\u2033}+(2x+3){y}^{\prime}+(x+3)y=3{e}^{-x}$$

Differential EquationsAnswered question

pin1ta4r3k7b 2022-11-07

Solve the ODE ${y}^{\u2033}+t{y}^{\prime}-y=0$ when y(0)=0 and y′(0)=5 by Laplace transform

My try:

$Y:=L\{y\}$

$s(sY-y(0))-5+L\{t{y}^{\prime}\}-Y=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{s}^{2}Y-5-(L\{{y}^{\prime}\}{)}^{\prime}-Y=0$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y({s}^{2}-1)-5-(Y+s{Y}^{\prime})=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y({s}^{2}-2)-5-s{Y}^{\prime}=0$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{Y}^{\prime}+\frac{2-{s}^{2}}{s}Y=\frac{-5}{s}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y=c{e}^{\frac{{s}^{2}}{2}}(\frac{1}{{s}^{2}})(1+5{e}^{-\frac{{s}^{2}}{2}})$

Now, the problem is that I don't know how to find the inverse Laplace of $Y=c{e}^{\frac{{s}^{2}}{2}}(\frac{1}{{s}^{2}})(1+5{e}^{-\frac{{s}^{2}}{2}})$

Help with it

My try:

$Y:=L\{y\}$

$s(sY-y(0))-5+L\{t{y}^{\prime}\}-Y=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{s}^{2}Y-5-(L\{{y}^{\prime}\}{)}^{\prime}-Y=0$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y({s}^{2}-1)-5-(Y+s{Y}^{\prime})=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y({s}^{2}-2)-5-s{Y}^{\prime}=0$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{Y}^{\prime}+\frac{2-{s}^{2}}{s}Y=\frac{-5}{s}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}Y=c{e}^{\frac{{s}^{2}}{2}}(\frac{1}{{s}^{2}})(1+5{e}^{-\frac{{s}^{2}}{2}})$

Now, the problem is that I don't know how to find the inverse Laplace of $Y=c{e}^{\frac{{s}^{2}}{2}}(\frac{1}{{s}^{2}})(1+5{e}^{-\frac{{s}^{2}}{2}})$

Help with it

Differential EquationsAnswered question

gfresh86iop 2022-11-07

How to find the inverse laplace transform of $$\frac{s(c-F(s))}{s-a}$$ where a , c are constants and ${L}^{-1}\{F(s)\}=f(t).$

Differential EquationsAnswered question

Messiah Sutton 2022-11-07

Let $\Vert \cdot \Vert $ be an arbitrary vector norm in ${\mathbb{C}}^{n}$. For matrix $A\in {\mathbb{C}}^{n}$, we define

$$\alpha (A)=\underset{\Vert x\Vert =1}{min}\Vert Ax\Vert .$$

Prove that if $\alpha (A)>0$ then A is invertible and $\Vert {A}^{-1}\Vert \le \frac{1}{\alpha (A)}.$

Can you give me hints on where to start?

$$\alpha (A)=\underset{\Vert x\Vert =1}{min}\Vert Ax\Vert .$$

Prove that if $\alpha (A)>0$ then A is invertible and $\Vert {A}^{-1}\Vert \le \frac{1}{\alpha (A)}.$

Can you give me hints on where to start?

Differential EquationsAnswered question

Ty Moore 2022-11-07

I'm currently studying transform of discontinuous and periodic functions

$$\frac{s{e}^{-3s}}{{s}^{2}+4s+5}$$

I've identified F(s) as:

$$\frac{s}{{s}^{2}+4s+5}$$

but I'm a little stuck on how to find the inverse Laplace transform of this. Do I complete the square?

$$\frac{s{e}^{-3s}}{{s}^{2}+4s+5}$$

I've identified F(s) as:

$$\frac{s}{{s}^{2}+4s+5}$$

but I'm a little stuck on how to find the inverse Laplace transform of this. Do I complete the square?

Differential EquationsAnswered question

sbrigynt7b 2022-11-07

How to calculate the Laplace transform

$$h:t\in [0,+\mathrm{\infty}[\to {\int}_{t}^{\mathrm{\infty}}\frac{1}{{e}^{s}\sqrt{s}}ds$$

I have to calculate the Laplace transform of h in 0

I know that $L[{\int}_{0}^{\mathrm{\infty}}f(t)dt](p)=\frac{1}{p}L[f](p)$ but i don't understand how to procede next.

$$h:t\in [0,+\mathrm{\infty}[\to {\int}_{t}^{\mathrm{\infty}}\frac{1}{{e}^{s}\sqrt{s}}ds$$

I have to calculate the Laplace transform of h in 0

I know that $L[{\int}_{0}^{\mathrm{\infty}}f(t)dt](p)=\frac{1}{p}L[f](p)$ but i don't understand how to procede next.

Differential EquationsAnswered question

perlejatyh8 2022-11-06

If $\mathcal{L}[f(t)]=\hat{f}(p)$If $\mathcal{L}[f(t)]=\hat{f}(p)$ then $\mathcal{L}[{e}^{at}f(t)]=\hat{f}(p+a)$

I have as far as $\mathcal{L}[{e}^{at}f(t)]={\int}_{0}^{\mathrm{\infty}}{e}^{-(s+a)t}f(t)dt$

How do I proceed?

I have as far as $\mathcal{L}[{e}^{at}f(t)]={\int}_{0}^{\mathrm{\infty}}{e}^{-(s+a)t}f(t)dt$

How do I proceed?

Differential EquationsAnswered question

bucstar11n0h 2022-11-06

Laplace transform of $f(t)=|\mathrm{sin}\frac{t}{2}|$?

how do you find the Laplace transform of this?

I tried using the equation

$$L\{f(t)\}=\frac{1}{1-{e}^{-sT}}{\int}_{0}^{T}f(t){e}^{-sT}\phantom{\rule{thinmathspace}{0ex}}dt$$

with period $T=2\pi $ but I am not sure if that's correct.

how do you find the Laplace transform of this?

I tried using the equation

$$L\{f(t)\}=\frac{1}{1-{e}^{-sT}}{\int}_{0}^{T}f(t){e}^{-sT}\phantom{\rule{thinmathspace}{0ex}}dt$$

with period $T=2\pi $ but I am not sure if that's correct.

Differential EquationsAnswered question

Paula Cameron 2022-11-06

What f(t) satisfies the inverse Laplace transform ${\mathcal{L}}^{-1}\left\{\frac{{p}^{\prime}(s)}{p(s)}\right\}=f(t),$ where the polynomial p is given.

Differential EquationsAnswered question

assupecoitteem81 2022-11-06

Prove the result of a Laplace transformation

$$\mathcal{L}({\int}_{0}^{t}\frac{1-{e}^{-u}}{u}du,s)=\frac{1}{s}\mathrm{log}(1+\frac{1}{s})$$

$$\mathcal{L}({\int}_{0}^{t}\frac{1-{e}^{-u}}{u}du,s)=\frac{1}{s}\mathrm{log}(1+\frac{1}{s})$$

Differential EquationsAnswered question

figoveck38 2022-11-06

If $H(s)=F(s)G(s)$ then

$$h(t)={L}^{-1}\{F(s)G(s)\}={\int}_{u=0}^{t}f(t-u)g(u)du$$

I want to know if

$$H(s)=F(P(s))G(P(s))$$

where P(s) is a function of s, then the inverse of H(s) i.e h(t) will be..........?

$$h(t)={L}^{-1}\{F(s)G(s)\}={\int}_{u=0}^{t}f(t-u)g(u)du$$

I want to know if

$$H(s)=F(P(s))G(P(s))$$

where P(s) is a function of s, then the inverse of H(s) i.e h(t) will be..........?

Differential EquationsAnswered question

Kayley Dickson 2022-11-06

Prove the Laplace transform equality

$${\mathcal{L}}^{-1}(\sqrt{s-\alpha}-\sqrt{s-\beta})=\frac{1}{2t\sqrt{\pi t}}[{e}^{(\beta t)}-{e}^{(\alpha t)}]$$

$${\mathcal{L}}^{-1}(\sqrt{s-\alpha}-\sqrt{s-\beta})=\frac{1}{2t\sqrt{\pi t}}[{e}^{(\beta t)}-{e}^{(\alpha t)}]$$

Differential EquationsAnswered question

Ty Moore 2022-11-06

I have a differential equation as such:

$$\frac{{d}^{2}i}{d{t}^{2}}+7\frac{di}{dt}+12i(t)=36\delta (t-2)$$

Where $i(0)=0,{i}^{\prime}(0)=0$. The solution is given by:

$$i(t)=u(t-2)[A{e}^{-3(t-2)}+B{e}^{-4(t-2)}]$$

Out of the solution, I need to express A and B as decimal numbers. Now, I can solve simpler differential equations, but the δ is completely throwing me off.

$$\frac{{d}^{2}i}{d{t}^{2}}+7\frac{di}{dt}+12i(t)=36\delta (t-2)$$

Where $i(0)=0,{i}^{\prime}(0)=0$. The solution is given by:

$$i(t)=u(t-2)[A{e}^{-3(t-2)}+B{e}^{-4(t-2)}]$$

Out of the solution, I need to express A and B as decimal numbers. Now, I can solve simpler differential equations, but the δ is completely throwing me off.

Differential EquationsAnswered question

inurbandojoa 2022-11-06

Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ?

A geometric Brownian motion $X=({X}_{t}{)}_{t\ge 0}$ satisifies $d{X}_{t}=\sigma {X}_{t}\phantom{\rule{thinmathspace}{0ex}}d{W}_{t}$ where $W=({W}_{t}{)}_{t\ge 0}$ denotes a Brownian motion and the associated integrated Brownian motion is ${\int}_{0}^{t}{X}_{s}\phantom{\rule{thinmathspace}{0ex}}ds$. The Laplace transform of an integrated gometric Brownian motion is thus

$$\mathcal{L}(\lambda )=\mathbb{E}\left[{e}^{-\lambda {\int}_{0}^{t}{X}_{s}ds}\right]$$

A geometric Brownian motion $X=({X}_{t}{)}_{t\ge 0}$ satisifies $d{X}_{t}=\sigma {X}_{t}\phantom{\rule{thinmathspace}{0ex}}d{W}_{t}$ where $W=({W}_{t}{)}_{t\ge 0}$ denotes a Brownian motion and the associated integrated Brownian motion is ${\int}_{0}^{t}{X}_{s}\phantom{\rule{thinmathspace}{0ex}}ds$. The Laplace transform of an integrated gometric Brownian motion is thus

$$\mathcal{L}(\lambda )=\mathbb{E}\left[{e}^{-\lambda {\int}_{0}^{t}{X}_{s}ds}\right]$$

Differential EquationsAnswered question

Aleah Avery 2022-11-06

Find a Laplace transform for the following function:

$$f(s)=\frac{s}{\sqrt{{a}^{2}-{\left(\frac{s}{2}\right)}^{2}}}$$

Does this function have a Laplace function for general a?

$$f(s)=\frac{s}{\sqrt{{a}^{2}-{\left(\frac{s}{2}\right)}^{2}}}$$

Does this function have a Laplace function for general a?

Differential EquationsAnswered question

Jaiden Elliott 2022-11-06

Find inverse laplace transform :

$$F(s)=\frac{2{\omega}^{3}}{({s}^{2}+{\omega}^{2}{)}^{2}}$$

$$F(s)=\frac{2{\omega}^{3}}{({s}^{2}+{\omega}^{2}{)}^{2}}$$

Differential EquationsAnswered question

Howard Nelson 2022-11-05

How to work out the Laplace transform with respect to t of:

$$\mathrm{sin}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}$$

I know that the transform of $\mathrm{sin}\left(t\right)$ is $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{a}{{s}^{2}+{a}^{2}}}$, and transform of $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}\phantom{\rule{thinmathspace}{0ex}}$ is ${s}^{2}F\left(s\right)-s\phantom{\rule{thinmathspace}{0ex}}f\left(0\right)-s\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime}\left(0\right)$

$$\mathrm{sin}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}$$

I know that the transform of $\mathrm{sin}\left(t\right)$ is $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{a}{{s}^{2}+{a}^{2}}}$, and transform of $\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{{d}^{2}y}{{dt}^{2}}}\phantom{\rule{thinmathspace}{0ex}}$ is ${s}^{2}F\left(s\right)-s\phantom{\rule{thinmathspace}{0ex}}f\left(0\right)-s\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime}\left(0\right)$

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.