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

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Are there functions that are not of exponential order for which you can define a Laplace transform?

I'am in a course of Introduction to Linear Differential Equations and teacher made us this question in class. we work in $\mathbb{R}$, and any help to answer this is welcome
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determine the time domain equation of the output responseusing inverse laplace transform, given a step input

I have this initial transfer function \begin{equation*} \frac{Y_s}{F_s}=\frac{1}{(1s^2+2s+2)} \end{equation*} unit step is $F_s=\frac{1}{s}$ so I then get \begin{equation*} \frac{1}{s(s^2+2s+2)} ...
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29 views

Find the roots of the corresponding characteristic equation

The equation is $${Y_s\over F_s}={1\over s^2+2s+2}$$ I have got to $$r^2+2r+2=0$$ what do i need to do next?
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46 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 ...
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37 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 ...
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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 ...
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65 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 ...
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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 ...
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24 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' + ...
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33 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,\ ...
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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 ...
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78 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 $x>x_0$, $y↦ F(x + iy )$ ...
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86 views

A difficult integral: 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 $$ ...
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63 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 ...
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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 ...
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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 ...
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36 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 ...
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26 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)} = ...
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30 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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64 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 ...
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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 ...
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Trouble with Laplace Tranform [closed]

Can anyone help me with this Laplace Transform $$\mathcal{L}[(1-\cos(u))/u] ?$$ Thanks in advance
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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 ...
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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?
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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) $? I have no idea where to start.
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$\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.
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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 ...
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$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$ - 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 ...
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61 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 ...
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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, ...
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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 ...
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49 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 ...
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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 ...
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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 $$\ddot{y}(t)+4\dot{y}(t)+5y(t)=u(t), \quad y(0)=0,\quad \dot{y}(t)=0,$$ ...
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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: $\mathcal{L}( \sin({3 t}) \cos({5 t}) )$
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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..
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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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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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?
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51 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 ...
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48 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 ...
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67 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 ...