The Laplace transform is a widely used integral transform, similar to the Fourier transform.

learn more… | top users | synonyms

0
votes
0answers
12 views

Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
1
vote
1answer
23 views

solving second order linear differential equation

Can somebody please show me how to solve the following differential equation: $$ a\ddot{x} + b\dot{x} = c $$ given these initial conditions $x(0) = 2$, $\dot{x}(0) = 0.5$ and $a = 4, b = 1.5$ First ...
0
votes
0answers
20 views

Laplace transform and “imaginary infinity”

I was recently studying Laplace transform for the first time, and I'd like to ask the following thing: there was an integral with limit of integration, something like that: a+j×infinity, j the ...
1
vote
1answer
23 views

Laplace transform involving the gamma function.

Does anyone know how to evaluate the following integral $$ \int_{0}^{\infty} \frac{e^{-qs}\alpha^{s}}{\Gamma(s)\Gamma(s)}\text{d}s $$ where $q,\,\alpha > 0$? I've done some digging in usual ...
0
votes
0answers
22 views

Laplace Transformation : differential equations

Dont know how to proceed further. Please Help me guys
0
votes
1answer
12 views

How can we make sure result of Laplace Transformation has no pole using lhopital's rule?

If there is $x(t) = rect(\frac{t}{2})$, then its L.T will be $X(s) = 1/s(e^s - e^{-s})$. right? and after that i tried to draw them on S-Plane to check if poles exist. In the L.T result, it looks like ...
1
vote
1answer
20 views

Solving IVP by Laplace transform

I'm trying to solve an IVP with non-constant coefficients $$ y'' + 3ty' - 6y = 1, \quad y(0) = 0, \; y'(0) = 0 $$ Taking the Laplace yields $$ s^2Y + 3(Y + sY') - 6Y = \frac{1}{s}$$ $$ Y' + ...
2
votes
1answer
19 views

Inverse Laplace Transform with time delay and extra factor

I am attempting to solve a PDE $$y_{tt} = y_{xx}, -\infty < x < 0,\ t > 0$$ with boundary conditions $$ y_x(0,t) = k(t),\ y(x,t) \rightarrow 0\ \mbox{as}\ x \rightarrow -\infty,\ y(x,0) = 0,\ ...
1
vote
2answers
29 views

Find the solution for the spring-mass problem $y′′+9y=\cos(3t)$. Solve with initial conditions $y(0) = 0$, $y′ (0) = 0$. Using Laplace transform

I first took the Laplace transform of each part then getting $s^{2}Y+9Y=\frac{s}{s^{2}+9}$ then solving for Y, I got $Y=\frac{s}{(s^{2}+9)^{2}}$ but don't know how to simplify that to something that ...
1
vote
1answer
38 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
2
votes
0answers
28 views

Laplace transform of Gaussian*Erfi

$$\sqrt{\frac{\pi }{2}} e^{-\frac{t^2}{2}} \text{erfi}\left(\frac{t}{\sqrt{2}}\right) \rightarrow -\frac{1}{2} e^{\frac{s^2}{2}} \text{Ei}\left(-\frac{s^2}{2}\right)$$ or $$ ...
1
vote
2answers
42 views

y''+xy'+y=0, y(0)=1, y'(0)=-1

I have used laplace transform to get $Y'(s)-sY'(s)=-1+\frac{1}{s}$ $Y(s)=-e^\frac{s^2}{2}\int e^\frac{-s^2}{2}ds + e^\frac{s^2}{2}\int \frac{ e^\frac{-s^2}{2}}{s}ds +Ce^\frac{s^2}{2}$ what should ...
0
votes
3answers
50 views

How to prove that $\int_{a}^{+\infty}\int_{0}^{+\infty}e^{-xt}\sin t\,dx\,dt = \int_{0}^{+\infty}\frac{\cos a+x\sin a}{1+x^2}e^{-ax}\,dx$

I want to prove $\int_{a}^{+\infty}\int_{0}^{+\infty}e^{-xt}\sin t\,dx\,dt = \int_{0}^{+\infty}\frac{\cos a+x\sin a}{1+x^2}e^{-ax}\,dx$. Should the proof be done using some kinds of Laplace transform ...
0
votes
0answers
22 views

Fourier transform from Laplace transform

So what I did was Laplace transform $f(t)$ to $F(s)$ and then plug in $s=jw$. However, when I tried this for $cos(3t)$, $$F(jw)={jw\over9-w^2}$$ was the answer. I don't know if this is correct, and ...
0
votes
1answer
32 views

How to solve this Inverse Laplace Transform

How would I solve this Inverse Laplace transform? $$\mathscr{L}_s^{-1} \left\{ \frac{s}{s^2-s+\frac{17}{4}} \right\}$$ The solution is $$f(t) = (1/4 )e^{t/2} (\sin(2 t)+4 \cos(2 t))$$ I know I need ...
0
votes
1answer
18 views

Is the Laplace Transform of the convolution power the product of the Laplace Transformed convolution?

In statistics, the definition of $F^k$ is the k-fold convolution of $F$ with itself, where $F$ is some common distribution. I am wondering if the following holds, if: $$ L_{F^{k}(x)} = ...
1
vote
0answers
23 views

Solve pde using laplace?

I have to solve the following pde using Laplace transforms: $xw_x + w_t= xt$ i.c: w(x,0)= 0 Firstly, transforming the above wrt t, i get: $\bar{w_x} + s\bar{w}/x = 1/s^2$ But, in the textbook, the ...
0
votes
1answer
17 views

How to solve for the inverse Laplace Transform

How would one solve the following inverse Laplace transform? $$\mathscr{L}_s^{-1}\left\{\frac{2s}{\left(s-1\right)^2+7}\right\}$$ I know from WolframAlpha that the answer is: $$\frac{2 e^t ...
2
votes
0answers
15 views

Asymptotics of Laplace transform at minus infinity

I am interested in relating the asymptotic behavior of a function $f(t)$ for large values of $t$ with the asymptotic behavior of its Laplace transform $\hat{f}(s)$ for small values of $s$. In practice ...
1
vote
0answers
17 views

How to apply the laplace transform to this second order ODE?

Can I apply the Laplace transform on a the following second order nonlinear PDE? $$ \frac{\partial y}{\partial t}=\frac{\partial^2 y^n}{\partial x^2}$$ where $n$ is a natural number? I mean apply the ...
0
votes
1answer
19 views

Utility of the Derivative of Laplace Transforms for ODE's

Many texts discuss the derivative of Laplace transform $dF(s)/ds$. In general, differentiation of a Laplace is equivalent to multiplying the original function by $t$, and vice versa. So, if ...
2
votes
0answers
14 views

For what types of differential equations is the Laplace transform most effective?

I'm reviewing for a final exam and want to make sure I know what tools to use for what situations, and was just wondering if there were situations where the Laplace transform is unusable or less ...
1
vote
1answer
39 views

Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
4
votes
0answers
32 views

Is the Laplace transform a vector space isomorphism? And what space is it isomorphic to?

The laplace transform is a linear transformation, $\mathcal{L}: \mathcal{M} \rightarrow?$, where $\mathcal{M}$ is the set of exponentially bounded functions on $\mathbb{R},$since ...
2
votes
2answers
24 views

Trouble with Laplace Tranform [closed]

Can anyone help me with this Laplace Transform $$\mathcal{L}[(1-\cos(u))/u] ?$$ Thanks in advance
1
vote
1answer
21 views

Chemical kinetics using Laplace transformation

I have a simple chemical reaction $A\leftrightarrow B$ with forward rate $k_1$ and backward rate $k_2$. I can now write the differential equation of this system as following. $ \frac{dA}{dt} = -k_1A ...
0
votes
1answer
11 views

How to find the inverse Laplace transform and solve for a?

The equation $\dfrac{Y(s)}{s^2} + \dfrac{Y'(s)}{s} = \dfrac{-a}{s^4}$ is in the Laplace transform. How can I take the inverse i.e transform back to time domain and solve for a?
4
votes
2answers
38 views

Finding the inverse Laplace transform of $ \ln \! \left( 1 + \frac{1}{s^{2}} \right) $.

Can someone help me find the inverse Laplace transform of $ \ln \! \left( 1 + \dfrac{1}{s^{2}} \right) $? I have no idea where to start.
0
votes
2answers
62 views

$\int^\infty_0 e^{-\alpha x}\sin(\beta x)\,dx = \frac{B}{\alpha^2+\beta^2}$ Laplace [closed]

$$ \int^\infty_0 \! e^{-\alpha x} \sin(\beta x)\,dx = \frac{\beta}{\alpha^2+\beta^2} $$ Can someone start this for me? I don't know where to begin.
1
vote
0answers
19 views

Show that $\forall n\in \mathbb{N}$, the funtion $e^{-x^n}$ is of exponential order and its Laplace transform exists on $[0,\infty)$

Show that $\forall n\in \mathbb{N}$, the funtion $e^{-x^n}$ is of exponential order and its Laplace transform exists on $[0,\infty)$ So we need to show that $e^{-sx} |f(x)|$ converges to show that it ...
3
votes
1answer
43 views

$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$ - different answers depending on value of $t$?

After taking an inverse Laplace transform I have the following - $$y = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$$ In my notes it says if $t ...
0
votes
1answer
22 views

Absolutely integrable function not of exponential order

Construct an example of a continuous function $y=f(x)$ defined on $[0,\infty)$, such that it is absolutely integrable, i.e., $\int^\infty_0 |f(x)|dx<\infty$, but not of exponential order. What ...
0
votes
0answers
9 views

if $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$

if laplace transform $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$. i need to prove this . now, ...
0
votes
0answers
20 views

A question of multi-dimensional integral

Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$ Is there a sufficiently condition on the ...
1
vote
1answer
40 views

Conditions for Laplace Transform

Consider the Laplace transform: $$\mathscr{L}(1) = \int_0^\infty e^{-st}dt = -\left.\frac{1}{s}e^{-st}\right|_0^\infty = \frac1s$$ Math textbooks usually state that this is only valid for the ...
0
votes
1answer
11 views

Question regarding $\mathcal{L}\{t*\mathcal{U}(t-2)\}$

I'm working on a problem for homework (* is multiplication not convolution): $\mathcal{L}\{t*\mathcal{U}(t-2)\}$ I understand that $\mathcal{L}\{(t-a)\mathcal{U}(t-a)\}=e^{-as}F(s)$ The first step ...
1
vote
1answer
18 views

Laplace transform using the convolution theorem

(This question is about laplace transforms) By making use of the convolution theorem show that the solution $y(t)$ to the ODE $$\ddot{y}(t)+4\dot{y}(t)+5y(t)=u(t), \quad y(0)=0,\quad \dot{y}(t)=0,$$ ...
0
votes
1answer
12 views

Laplace transform of multiplication of two terms with different arguments

What is the Laplace Transform of the product of two functions with different arguments? The function is: $\mathcal{L}( \sin({3 t}) \cos({5 t}) )$
0
votes
0answers
21 views

Doubt in laplace transforms

Let $f(t)=e^{t^2}$. Now the laplace transform of $f(t)$ is $$\int_0^\infty e^{-st}e^{t^2}dt=\int_0^\infty e^{-st+t^2}dt$$ But after this.. How can I proceed? Help me..
0
votes
0answers
11 views

Finding a basic laplace transform

Find the laplace transform of the function $f(t)=t^3e^{4t}$. The solution I am presented is Now $\mathcal{L}(e^{4t}f(t)) = F(s-4)$ and $\mathcal{L}(t^3) = 6/s^4$. So the Laplace transform of $f$ is ...
1
vote
0answers
19 views

Solving IVP using Laplace Transform

Let $$g(t) =\begin{cases} t & \text{if $t \leq6π$} \\ 6\pi & \text{if $t>6\pi$} \end{cases} $$ Solve $y''+ 16y = g(t)$ where $y(0) = 9$ and $y'(0) = 4$ using Laplace transforms. I got ...
2
votes
1answer
20 views

Question regarding $\mathcal{L}^{-1}\{\frac{s}{s^2+4s+5}\}$

The book asks for: $\mathcal{L}^{-1}\{\frac{s}{s^2+4s+5}\}$ So I can see: $\frac{s}{s^2+4s+4+1} = \frac{s}{(s+2)^2+1}$ From the properties of the inverse Laplace transform: ...
8
votes
1answer
79 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
1
vote
2answers
22 views

What is my error in applying this Laplace Transform?

So, our book has the seemingly innocuous problem: $y''-y'-6y=0$. I was able to solve by hand, and come up with $${\scr L}(y)=\frac{s-2}{s^{2}-s+6}$$.That completed, I factored the bottom to ...
0
votes
0answers
28 views

Is $\cosh(t^2)$ of exponential order?

Is $\cosh(t^2)$ of exponential order? I know that it isn't, but I am unsure as to why. Also why is $\cosh(t) $ of exponential order?
0
votes
1answer
31 views

Solving 2nd Order ODE w/Laplace Transforms + Heaviside

This is a similar problem to the one I posted earlier with some differences. Attempt at solution: Write g(t) as a heaviside function. Take Laplace transform of LHS and RHS. Solve for Y. Take ...
0
votes
1answer
38 views

Solving 2nd Order ODE w/Laplace Transforms

I am having difficulty with this problem: *Note: The Delta3(t) is the delta dirac function, also the answer in the image is WRONG. Attempt at solution : Let Laplace{y(t)}=Y Take Laplace of LHS ...
1
vote
1answer
31 views

Laplace transform nonlinear equation

How can I apply the Laplace transform on a the following nonlinear PDE $$ \frac{\partial y}{\partial t}=\frac{\partial y^n}{\partial x}$$ where $n$ is a natural number? When I say apply the Laplace ...
1
vote
1answer
19 views

Find the distribution with the following Laplace transform.

Is anybody aware of the distribution whose Laplace transform is the following. \begin{equation} \mathbb{E}[e^{-tX}] = \frac{e^{-t}}{(1+2t)} \end{equation} Note: The Laplace transform of the ...
2
votes
2answers
36 views

Inverse Laplace Transform of $\ln[\frac{s^2+a^2}{s^2+b^2}]$

How does one find $\mathcal{L}^{-1}\{\ln[\frac{s^2+a^2}{s^2+b^2}]\}$? I've tried splitting it up into $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}-\mathcal{L}^{-1}\{\ln(s^2+b^2)\}$. However, I can't think of ...