2
votes
1answer
27 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
0
votes
1answer
45 views

Solving and graphing an IVP involing unit step function

Im trying to solve this ODE and find a simplified expression for $x(t)$. $$\ddot x+4x=-2\sum_{n=1}^{4} e^{in\pi}u(t-n\pi);\space x(0)=0=\dot x(0),i=\sqrt{-1}$$ First i found the the laplace ...
0
votes
2answers
25 views

Convolution, indicator function

I need to calculate $(f*f)(x)$ of $f(x) = 1_{[0,1]}(x)$, which is the indicator function defined with Calculating the integral $(f*f)(x) = \int_{0,}^{x}1_{[0,1]}(t) \cdot1_{[0,1]}(x-t) dt$ gives ...
0
votes
1answer
50 views

Laplace transform of unit step function

Im given a graph of $f(t)$ and i need to find the Laplace transform of $f(t)$. From looking at the graph i have $$f(t) = \begin{cases} t, & \text{$0 \le t \le 1 $} \\ 0, & \text{$1 \lt t \lt ...
0
votes
1answer
35 views

Laplace transform of initial value problem, stuck on partial fractions.

The problem im given is: Use Laplace transforms to solve the initial value problem. $$\ddot x +x=\sin(2t)$$ $$x(0)=0=\dot x(0)$$ I first do the following Laplace transforms: $$\mathcal{L}\{\ ...
2
votes
2answers
60 views

Laplace transform with initial value problem $y''+4y=12\sin(2t)$.

Using Laplace transforms solve the initial value problem. $$y''+4y = 12\text{sin}(2t); \qquad\qquad y(\pi)=-3, \quad y'(\pi)=-3$$ I have begun with writing: $\mathcal{L} (y'') = s^2y(s) -s y(\pi) ...
1
vote
1answer
27 views

Laplace transform of piecewise continuous function

$$f(t) =\begin{cases}t^2 & 0 \le t < 3,\\ 9& t \ge 3\end{cases}$$ Show that $f$ is of exponential order. Express $f$ in terms of the unit step function. Find Laplace transform of ...
1
vote
1answer
34 views

Laplace transform of convolution with no function of t

Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming. Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$ My Problem: To treat this as ...
0
votes
0answers
83 views

Proof of modifed Final Value theorem

Let $f:[0,\infty] \longrightarrow C $ be a continuous and bounded function such that the limit $ \lim_{t\to \infty } \frac1T \int_0^T f(t) dt = d $ exists. Let $ F(s) $ be the Laplace transform ...
1
vote
2answers
129 views

Solve second order differential equation with Heaviside function using Laplace transform

The equation is: $$y'' + 3y = u_4(t)\cos(5(t-4)), \quad y(0) = 0, \quad y'(0) = -2$$ Here $u_4$ is the Heaviside function with activation switch at $t=4$. I can get all the way to the partial ...
0
votes
0answers
56 views

Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
0
votes
3answers
88 views

Coupled mass spring system with damping, I need help with the equation

I know that the equation $mx''+cx'+kx=f(t)$ is used for a normal mass spring system, but I don't know how to express the differential equation for a coupled mass spring system with damping. These are ...
1
vote
1answer
41 views

Circuit RC, I need help with the equation.

A circuit RC it's described by the next equation: $\frac{1}{c} \int i(dt)+Ri=V$ Where the value of resistance is $R=10 k\omega $ , the value of the capacitor is $C=2.5 \mu F$, and the voltage of the ...
0
votes
1answer
36 views

Inverse Laplace Transform, I need help

What is the ILT of $H(s)=\frac{7(3s+1)}{(s-3)(s^2+10s-13)}$ Also, if you kindly want to help with this another inverse transform, I'd really appreciate it: $H(s)=\frac{6(s+2)}{s^3(s-5)}$ Thanks!
1
vote
1answer
78 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
0
votes
1answer
85 views

Simple inverse using Laplace transform

I have the following excercise. I looked at the Laplace transform table for said transform, but I can't find any that looks similar to this. Help, please? $$ \mathcal{L} ^ {-1} \left[ \frac{s} {((s + ...
0
votes
1answer
37 views

Simple inverse laplace transform problem

I have the follow excercise. I am aware of partial fraction expansion, but the roots are imaginary in this problem. Does somebody know how to solve it? Thanks. $$ \mathcal{L} ^ {-1} (1 / (s ^ 2 + 4 s ...
0
votes
1answer
55 views

Laplace Transformations of $\frac{1}{t}$

what is the laplace transform of $\frac{1}{t}$? I tried different ways like integrating by parts from the general form of laplace but it's getting more complex as my solution goes by.
1
vote
1answer
37 views

Find the Laplace transform of the following hard equation

Ok so the objective is to factor this into something that resembles the Laplace tables. Give me some help pls. thx
1
vote
2answers
215 views

Laplace transform with time shift property

ok so i have no idea how the inverse laplace went from $F(s)$ to $f(t)$. I understand $\frac{c}{s^2}$ => $ct$, and $\frac{b}{s}$ => $b$, but the $e^{-as}$ is what gets me. In my Laplace tables I ...
1
vote
1answer
60 views

Laplace question

How do you express Laplace transform $\mathcal{L}(g)(z)=\int_{0}^\infty e^{-zt}g(t)dt$ with Fourier transform? And how do you form the reverse formula for Laplace transform using Laplace transform ...
-2
votes
1answer
57 views

Find the inverse Laplace transformation of $\frac{(s+1)e^{-s}}{s^2}$. [closed]

Find the inverse Laplace transformation of $\dfrac{(s+1)e^{-s}}{s^2}$.
0
votes
3answers
61 views

$\mathcal {L}(\operatorname e^{-6t}\cos(5t))=?$ [closed]

Find the laplace transform of the following equation $f(t) =\operatorname e^{-6t}\cos (5t)$
0
votes
1answer
44 views

A physical system is found to have the following differential equation with all initial condition being zero [closed]

A physical system is found to have the following differential equation with all initial condition being zero $$ {{\rm d}^{2}{\rm X}\left(t\right) \over {\rm d}t^{2}} + 4\,{{\rm d}{\rm ...
2
votes
3answers
263 views

Solve the following with Laplace transform $y''-y'=f(t)$

$$\begin{cases} y''-y'=f(t) \\ y(0)=1 \\ y'(0)=1\\ \end{cases} $$ $f(t)$ is $$f(t)= \begin{cases} -1 & 0\leq t < \pi \\ cos(t) & \pi\leq t < 2\pi \\ ...
2
votes
2answers
31 views

Laplace transform of $(t-2)^2u_2$

The homework problem is $$f(t) = \begin{cases} 0 & t < 2\\ (t-2)^2 & t\geq 2\end{cases}$$ $f(t)$ as a step function $$f(t) = (t-2)^2u_2(t)$$ Using what we learned in class ...
2
votes
2answers
143 views

Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
0
votes
1answer
83 views

$f(t)=1+t-\dfrac{8}{3}\displaystyle\int_{0}^{t}(\tau-t)^3f(\tau) \ \mathrm d\tau \quad , f(t)=?$

$$ f(t)=1+t-\dfrac{8}{3}\displaystyle\int_{0}^{t}(\tau- t)^3f(\tau) \ \mathrm d\tau $$ According to the convolution theorem, $\displaystyle\int_{0}^{t}(\tau- t)^3f(\tau)d\tau$ = $f(t) * t^3$ (I ...
2
votes
1answer
48 views

What is the easiest way to find the inverse Laplace of F(s)?

$$ F(s)= \frac{1}{(s-1)^2(1-1/s^2)} $$ Do I have to multiply by $s^2/s^2$ and then use partial fractions or is there a way to use the convolution theorem?
1
vote
2answers
36 views

How do i find the lapalace transorm of this intergral using the convolution theorem?

$$\int_0^{t} e^{-x}\cos x \, dx$$ In the book, the $x$ is written as the greek letter "tau". Anyway, I'm confused about how to deal with this problem because the $f(t)$ is clearly $\cos t$, but ...
1
vote
1answer
235 views

Solve for z(t) from the simultaneous equation using Laplace transform

Solve for $z(t)$ from the simultaneous equation using Laplace transform $$ y' + 2y + 6 \int\limits_0^t z \mathrm{d}t = -2 u(t) \\ y' + z' + z = 0$$ subject to $y(0) = -5$ and $z(0) = 6$.
0
votes
1answer
67 views

determine the locations of all the poles and zeros (including zeros at s = infinite). Make an S-Plane plot of the infinite poles and zeros

Determine the locations of all the poles and zeros (including zeros at $S = \infty$). Make an $S$-Plane plot of the infinite poles and zeros. $$G(s) = \dfrac{5S^2 + 20S + 15}{S(S + 3)(S^2 + 4S + 4)}$$ ...
1
vote
3answers
71 views

How do i find the inverse laplace?

$$ F(s) = \frac{2s-1}{s^2(s+1)^3} $$ If I try to use partial fractions, I end up with 8 constants to solve for! Is there some shortcut I'm not seeing? Am I supposed to simplify it first? Am I even ...
1
vote
1answer
69 views

Laplace Transform using t-shift

$$f(t)=\begin{cases}cos(πt), & 1\leq t < 4 \\ 0, &elsewhere \end{cases}$$ Okay, I attempted to write it in terms of step functions and I got $$ f(t) = cos(πt)u(t-1)-cos(πt)u(t-4)$$ But ...
0
votes
1answer
81 views

Laplace Transform using t-shift (second shift)

$$f(t) = tu(t-π)$$ I know I have to get t in terms of $$(t-π)$$ and to do that I have done $$ t = a(t-π) + b$$ $$ t = at-aπ + b$$ $$ t = (a-π)t + b$$ $$ (a-π) = 1$$ and $$b = 0$$ Then I think I ...
-1
votes
1answer
244 views

How to use Complex Inversion Theorem to find the Inverse Laplace Transform?

How to use Complex Inversion Theorem to find the Inverse Laplace Transform for the given $F(t)=L^{-1} \{s^{-1/2} e^{-1/s}\}$ ? Hint: make the radius $\epsilon$ of the inner circle $t-1/2$ rather than ...
2
votes
2answers
414 views

Finding Inverse Laplace Transform using Taylor Series

Find the inverse Laplace transform $F(t)=\mathcal{L}^{-1}(s^{-\frac{1}{2}}e^{-\frac{1}{s}})$ using each of the following techniques: Expand the exponential in a Taylor series about s=∞, and take ...
1
vote
1answer
57 views

Computing the inverse Laplace transform of this?

What's the correct way to go about computing the Inverse Laplace transform of this? $$\frac{-2s + 1}{(s^2+2s+5)}$$ I Completed the square on the bottom but what do you do now? $$\frac{-2s + ...
1
vote
1answer
71 views

Laplace transformation problem

There is a timely unchanged continuous function : $$H(s)=\frac{s-1}{s+1}$$ At the entry of the system exists a $x(t)$ which Laplace's transformation is: $$X(s)=\frac{(5s^2 - 15s + ...
3
votes
2answers
237 views

Use Laplace transform to solve the following initial–value problems.

Use Laplace transform to solve the following initial–value problem. $y′′′′ + 2y′′ + y = 0, y(0) = 1, y′(0) = −1, y′′(0) = 0, y′′′(0) = 2$ Answer $s^4 L(s) - s^3y(0) -s^2 y'(0) - s y''(0) - y'''(0) ...
1
vote
1answer
107 views

Solving initial value problem using Laplace Transform

Use Laplace transform to solve the following initial–value problems. a). $y'' + y = e^{−t}\cos 2t, \\ y(0) = 2, y′(0) = 1$ After using the concept of partial fraction and using Elementary Laplace ...
0
votes
2answers
87 views

Laplace question continued (partial fractions)

Last night I attempted and successfully finished (with the help of stackexchange) the first part to this question on laplace transformations: Laplace question - help needed The second part to this ...
1
vote
2answers
80 views

Laplace transform of $|\sin(t)|$

There's already an answer to this, but I'm curious as to why my method of solving doesn't work. I take the integral where $\sin{t}$ is positive and the negative integral where it is negative: ...
1
vote
1answer
100 views

Differential equations - show solution given by expression

Show that the solution of the problem of Cauchy $\ddot{x}(t)=-a^2x(t)+ b(t)$ with $ x(0)=x_0$ and $\dot{x}(0)=v_0$ Is given by $\displaystyle x(t)=x_0 \cos(at)+\frac{v_0}{a} ...
1
vote
2answers
129 views

inverse of laplace transform

How to compute this inverse Laplace transform ? $$\displaystyle{ \mathcal{L^{-1}} \left\{ \frac{1}{s(\exp(s)+1)} \right\} }$$ Thanks.
-3
votes
1answer
1k views

Find the Laplace transform from given graph

![an image of the fuction][1] How do I find the function from its graph here to find its Laplace transform?
1
vote
2answers
122 views

How do I find Laplace transform of $7t \cdot \mathrm{e}^{-3t}\cdot\sin(3t)$?

I understand that the definition of a Laplace transform of a function $f(t)$ is $$F(s)= \int_0^{+\infty} e^{-st}f(t) \,dt$$ Is there an easy way to find the Laplace transform of $$f(t)=7t \cdot ...
2
votes
2answers
2k views

Finding the Laplace Transform of sin(t)/t

I'm in a Differential Equations class, and I'm having trouble solving a Laplace Transformation problem. This is the problem: Consider the function $$f(t) = \{\begin{align}&\frac{\sin(t)}{t} ...
1
vote
1answer
4k views

Inverse Laplace Transform of s/(s+1)

What is the inverse laplace transform of $\frac{s}{s+1}$? My work was: $$ X(s)=\frac{s}{s+1}\\ X(s)=s\frac{1}{s+1}\\ x(t)=\frac{d}{dt}e^{-t}=-e^{-t} $$ My only issue is that when I check my answer ...
0
votes
3answers
929 views

Laplace transform of $\cos(at)$

I need to find the Laplace transform of $\cos(at)$ I know that $L\{\cos(at)\}= \int_{0}^{\infty} e^{-st} \cos (at) dt$ but I am having trouble finding the integral Thank you