# Learn Laplace Transform Equations with Plainmath

Recent questions in Laplace transform
Taylor Barron 2022-11-08

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

Kayden Mills 2022-11-07

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

MMDCCC50m 2022-11-07

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

pin1ta4r3k7b 2022-11-07

## Solve the ODE ${y}^{″}+t{y}^{\prime }-y=0$ when y(0)=0 and y′(0)=5 by Laplace transformMy try:$Y:=L\left\{y\right\}$$s\left(sY-y\left(0\right)\right)-5+L\left\{t{y}^{\prime }\right\}-Y=0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}{s}^{2}Y-5-\left(L\left\{{y}^{\prime }\right\}{\right)}^{\prime }-Y=0$$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}Y\left({s}^{2}-1\right)-5-\left(Y+s{Y}^{\prime }\right)=0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}Y\left({s}^{2}-2\right)-5-s{Y}^{\prime }=0$$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}{Y}^{\prime }+\frac{2-{s}^{2}}{s}Y=\frac{-5}{s}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}Y=c{e}^{\frac{{s}^{2}}{2}}\left(\frac{1}{{s}^{2}}\right)\left(1+5{e}^{-\frac{{s}^{2}}{2}}\right)$Now, the problem is that I don't know how to find the inverse Laplace of $Y=c{e}^{\frac{{s}^{2}}{2}}\left(\frac{1}{{s}^{2}}\right)\left(1+5{e}^{-\frac{{s}^{2}}{2}}\right)$Help with it

gfresh86iop 2022-11-07

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

Messiah Sutton 2022-11-07

## Let $‖\cdot ‖$ be an arbitrary vector norm in ${\mathbb{C}}^{n}$. For matrix $A\in {\mathbb{C}}^{n}$, we define$\alpha \left(A\right)=\underset{‖x‖=1}{min}‖Ax‖.$Prove that if $\alpha \left(A\right)>0$ then A is invertible and $‖{A}^{-1}‖\le \frac{1}{\alpha \left(A\right)}.$Can you give me hints on where to start?

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?

sbrigynt7b 2022-11-07

## How to calculate the Laplace transform $h:t\in \left[0,+\mathrm{\infty }\left[\to {\int }_{t}^{\mathrm{\infty }}\frac{1}{{e}^{s}\sqrt{s}}ds$I have to calculate the Laplace transform of h in 0I know that $L\left[{\int }_{0}^{\mathrm{\infty }}f\left(t\right)dt\right]\left(p\right)=\frac{1}{p}L\left[f\right]\left(p\right)$ but i don't understand how to procede next.

perlejatyh8 2022-11-06

## If $\mathcal{L}\left[f\left(t\right)\right]=\stackrel{^}{f}\left(p\right)$If $\mathcal{L}\left[f\left(t\right)\right]=\stackrel{^}{f}\left(p\right)$ then $\mathcal{L}\left[{e}^{at}f\left(t\right)\right]=\stackrel{^}{f}\left(p+a\right)$I have as far as $\mathcal{L}\left[{e}^{at}f\left(t\right)\right]={\int }_{0}^{\mathrm{\infty }}{e}^{-\left(s+a\right)t}f\left(t\right)dt$How do I proceed?

bucstar11n0h 2022-11-06

## Laplace transform of $f\left(t\right)=|\mathrm{sin}\frac{t}{2}|$?how do you find the Laplace transform of this?I tried using the equation$L\left\{f\left(t\right)\right\}=\frac{1}{1-{e}^{-sT}}{\int }_{0}^{T}f\left(t\right){e}^{-sT}\phantom{\rule{thinmathspace}{0ex}}dt$with period $T=2\pi$ but I am not sure if that's correct.

Paula Cameron 2022-11-06

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

assupecoitteem81 2022-11-06

## Prove the result of a Laplace transformation$\mathcal{L}\left({\int }_{0}^{t}\frac{1-{e}^{-u}}{u}du,s\right)=\frac{1}{s}\mathrm{log}\left(1+\frac{1}{s}\right)$

figoveck38 2022-11-06

## If $H\left(s\right)=F\left(s\right)G\left(s\right)$ then$h\left(t\right)={L}^{-1}\left\{F\left(s\right)G\left(s\right)\right\}={\int }_{u=0}^{t}f\left(t-u\right)g\left(u\right)du$I want to know if$H\left(s\right)=F\left(P\left(s\right)\right)G\left(P\left(s\right)\right)$where P(s) is a function of s, then the inverse of H(s) i.e h(t) will be..........?

Kayley Dickson 2022-11-06

## Prove the Laplace transform equality${\mathcal{L}}^{-1}\left(\sqrt{s-\alpha }-\sqrt{s-\beta }\right)=\frac{1}{2t\sqrt{\pi t}}\left[{e}^{\left(\beta t\right)}-{e}^{\left(\alpha t\right)}\right]$

Ty Moore 2022-11-06

## I have a differential equation as such:$\frac{{d}^{2}i}{d{t}^{2}}+7\frac{di}{dt}+12i\left(t\right)=36\delta \left(t-2\right)$Where $i\left(0\right)=0,{i}^{\prime }\left(0\right)=0$. The solution is given by:$i\left(t\right)=u\left(t-2\right)\left[A{e}^{-3\left(t-2\right)}+B{e}^{-4\left(t-2\right)}\right]$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.

inurbandojoa 2022-11-06

## Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ?A geometric Brownian motion $X=\left({X}_{t}{\right)}_{t\ge 0}$ satisifies $d{X}_{t}=\sigma {X}_{t}\phantom{\rule{thinmathspace}{0ex}}d{W}_{t}$ where $W=\left({W}_{t}{\right)}_{t\ge 0}$ 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$\mathcal{L}\left(\lambda \right)=\mathbb{E}\left[{e}^{-\lambda {\int }_{0}^{t}{X}_{s}ds}\right]$

Aleah Avery 2022-11-06

## Find a Laplace transform for the following function:$f\left(s\right)=\frac{s}{\sqrt{{a}^{2}-{\left(\frac{s}{2}\right)}^{2}}}$Does this function have a Laplace function for general a?

Jaiden Elliott 2022-11-06