The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

learn more… | top users | synonyms

1
vote
0answers
38 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
54 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
51 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
36 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
27 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
42 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
26 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
30 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
19 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 ...
0
votes
1answer
137 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 ...
5
votes
0answers
166 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
28 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
244 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 ...
1
vote
2answers
27 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
3answers
438 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.
-1
votes
2answers
159 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
25 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
46 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
182 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 ...
1
vote
1answer
62 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
12 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
29 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
25 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
37 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..
1
vote
0answers
29 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
24 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
125 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
23 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
72 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
93 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 ...
1
vote
1answer
55 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
251 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
33 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
49 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 ...
0
votes
0answers
69 views

How to find the Laplace Transform of two (independent) functions multiplied together?

How does one find the laplace transform for an equation consisting of two trig functions multiplied together, when it is not possible to use any trig identities? For example, take a function say; ...
3
votes
3answers
204 views

Laplace transforms for a pharmacokinetics multi-compartmental model

I am an anaesthetist trying to write some pharmacokinetics software as a pet project. Unfortunately the maths I need is a bit too much for my rusty high school calculus, and I am out of my depth. I am ...
2
votes
2answers
28 views

Laplace inverse for Taylor expansion

By using infinite series find Laplace inverse for |1/(S^3+1)| .... I don't know what to do after using taylor expansion.. when I use it I got polynomial of $ S $ in the nominator which I can not deal ...
1
vote
1answer
21 views

The density of the distribution whose Laplace transform is the following

Is anybody aware of the density of the distribution whose Laplace transform is the following. \begin{equation} \mathbb{E}[e^{tX}] = \frac{e^{t/2}-1}{t/2} \end{equation} Note: $X$ is a continuous ...
1
vote
0answers
26 views

Why does the laplace transform of sine and cosine looks the way they are

I always forget what the Laplace transform of sine and cosine looks like. This is because they look so similar to each other. Does there exist a good mnemonic for remembering what form each takes? Or ...
2
votes
2answers
58 views

Laplace Transform of $\cosh(bt)$

So, one of my homework assignments is to take the Laplace transform of a function such that $f(t)=\cosh{bt}$. I figured it would be equivalent to: ...
0
votes
1answer
46 views

What is the General procedure for graphing heavidside functions?

I was given an example of a second order differential equation with U1(t)-U(3t) as the forcing function. I was asked to graph the forcing function and the answer is that the function is 1 when t is ...
0
votes
1answer
41 views

Inverse Laplace transform and convolution

Suppose we have two functions of $t$, $f(t)$ and $g(t)$. Letting $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$, I know that: $\mathcal{L}\{f(t) \star g(t)\} = F(s) \cdot G(s)$, but ...
3
votes
1answer
68 views

Finding the Laplace Transform Inverse

Solve by Laplace Transforms. So I'm stuck on how to find this $\mathcal{L}^{-1}$ $( \frac{\frac{5s}{4} + \frac{13}{4}}{s^2+5s+8} ) $ I'm not sure what t odo. I was thinking I need to use the ...
5
votes
1answer
274 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
2
votes
0answers
125 views

Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
0
votes
1answer
95 views

Final value theorem for closed system

We have a system with output given by $\frac{Y(s)}{R(s)} = \frac{F(s)G(s)}{1+F(s)G(s)}$ where $F(s)G(s) = K\frac{s+1}{s^2+s+1}$. Let $K=4$ and $R(s) = 10/s$. Using the final value theorem, ...
0
votes
2answers
79 views

solve $y''-4y=\delta(t-1)$ with initial conditions $y(0)=0, \; y'(0)=1$ using Laplace transforms

I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ ...
1
vote
1answer
43 views

Checking region of validity for a standard Laplace transform

In these notes, http://www.math.psu.edu/papikian/Kreh.pdf, Theorem 2.14 it states that $$\mathcal{L}[J_0](s)=\frac{1}{\sqrt{1+s^2}} $$ which I suppose is equivalent to $$ ...
0
votes
1answer
62 views

Laplace Transform to solve $R\frac{dQ}{dt}+\frac{Q(t)}{C}=V(t)$

I have the differential equation $R\frac{dQ}{dt}+\frac{Q(t)}{C}=V(t)$ where $R,C\in\mathbb R$ and $Q,V$ are functions of $t$. If I take the laplace transform of the differential equations I get: ...
1
vote
1answer
67 views

Final value of 1/(( s+2 )² * (s² - s + 1)) in the time domain

The original question is given as $$\frac {d^3y}{dt^3}+y=u=(1-t)e^{-2t}$$ The initial value y(0) = 0 and the same for all derivatives of y. Determine Y(s) What happens to u(t) and y(t) when ...