1
vote
1answer
27 views

the jump in $\ddot y$, Laplace transform

Given the following IVP: $$\ddot y+4y=\cos t-\cos t \cdot \theta(t-2\pi), y(0)=0, \dot y(0)=1$$ Check that $y(t)$ is continuous at $t=2\pi$. Find the jump in $\ddot y(t)$ at $t=2\pi$ i.e find $\lim ...
1
vote
1answer
18 views

Convolution and Total Response Differential Equations

Convolution with differential equations is extremely confusing to me. The two following questions were asked in class and we were asked to think about them. I want to work them out but I don't know ...
0
votes
1answer
31 views

ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
4
votes
2answers
41 views

Differential Equations with Discontinuous Forcing Functions

$$ y''+y'+1.25y = g(t), \quad t > 0, $$ $$y(0) = 0, \quad y'(0) = 0 $$ $$g(t) = \left\{ \begin{array}{ll} \sin{t} & 0 \le t < \pi \\ 0 & t \ge \pi \end{array}\right.$$ ...
-1
votes
1answer
56 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
0
votes
0answers
45 views

Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...
2
votes
2answers
94 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
1
vote
0answers
24 views

Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
2
votes
1answer
31 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 ...
5
votes
1answer
59 views

Laplace transformation $y''+2y'+2y=3\sin x+\cos x$

Given$$y''+2y'+2y=3\sin x+\cos x$$ Transform to image region $$Y(s)(s^2+2s+2)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ $$Y(s)((s^2+2s+1)+1)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ ...
0
votes
1answer
42 views

Laplace transform of a differential equation

Given the Laplace transform \begin{align} \mathcal{L}\{g(r)\} = f(t) = \int_{0}^{\infty} e^{-tr} g(r) \ dr \end{align} can it be shown that the transform of the differential equation \begin{align} ...
1
vote
3answers
60 views

Solving an ODE using Laplace Transforms

$$y′′′′ + 2y′′ + y = \sin x$$ $$y(0) = y′(0) = y′′(0) = y′′′(0)= 0$$ After solving I got $y(s)=\dfrac1{(s^2 + 1)^3}$ for which I am unable to find the inverse Laplace transform. Please let me know ...
0
votes
0answers
22 views

How to compute transfer function from Laplace Transform

My system of interest has the following EOM (V is my input variable): $\ddot{x} = g - k_{1}V(t) + \dot{x}k_2$ Taking the Laplace with initial conditions of zero, I get: $s^2X(s) = \frac{g}{s} - ...
1
vote
3answers
54 views

Inverse Laplace Transform,

I have been stuck on this problem for quite a bit, have tried to look at similar answers on website but no help... The original questions is, Solve the IVP $\ y''+y=\sin(t);y(0)=1;y'(0)=0$ I ...
1
vote
0answers
38 views

Laplace transform of $\sin(x)$

I am confused with Laplace transform of $\sin(\theta)$. For example, what is the LT of $A \sin(x(t))=Bx''(t)$ ($x$ is second order), $A,B$ are constants.
0
votes
0answers
52 views

solving differential equations with function coefficients using Laplace Transform

Does there exits a method to solve an $n$-th order liner differential equation with "function coefficients" using Laplace transform. It is well known that the identity $$L\left\{ {{t^n}f\left( t ...
0
votes
1answer
47 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 ...
1
vote
1answer
112 views

deriving second order transfer function from spring mass damper system..

I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...
0
votes
2answers
48 views

Unit Step Function

Question: What is $\mathcal{L}\{u(t-1)u(t-2)\}$? My calculations $e^{-2}s \mathcal{L}\{u(t+2)-1\}$ $e^{-2}s \mathcal{L}\{t+1\}$ $e^{-2}s (s^{-2}+\frac{1}{2})$ I'm confused, I gotten the wrong ...
0
votes
1answer
51 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 ...
2
votes
0answers
64 views

Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
0
votes
1answer
39 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}\{\ ...
0
votes
1answer
31 views

Using Laplace Transform to solve a 3 by 3 system of differential equations

I have been trying to solve this system of equations using Laplace transforms for a while. It is very easy to solve it using eigenvalues and eigenvectors, but when I tried to do it using Laplace I ...
0
votes
1answer
27 views

Differential Question about Laplace/Delta/Convolution

I need help understanding a part of this question. Let $a.) y''+4y = \delta (x)$, $y(0)=y'(0)=0$. and $b.) y'' + 4y = f(x)$, $y(0)=y'(0)=0$ where $f(x)$ is some continuous function of finite ...
1
vote
2answers
69 views

Solving a differential equation using the laplace transform involving convolution

The problem is the following The thing that puzzles me here is the integral on the right hand side, so: How to take the laplace transform on the right hand side? Any help to get me going would be ...
1
vote
2answers
38 views

integro-differential equation with application in quantum mechanics

I am trying to solve for the time dynamics for a simple quantum system (two-site system with sinusoidal coupling and a decay parameter on one site) and the math is looking not so simple. Here is the ...
2
votes
1answer
76 views

Solving a differential equation using Laplace transform

The problem has two parts: 1. Solve the initial value problem: $$ y''+y=\sum_{j=0}^\infty \delta_{2j\pi}(t) $$ with the initial conditions: $y(0)=y'(0)=0$ 2.Show that if $2n\pi<t<2(n+1)\pi$ ...
2
votes
1answer
55 views

Laplace Transform Piecewise Function

I've never seen these types of bounds on a piecewise function of a Laplace transform before, can someone help explain how to solve this problem, particularly the Laplace transform of g(t)? Thanks in ...
0
votes
0answers
34 views

What Laplace transformation is used for this?

I cannot see the transition here. I was unable to find what Laplace transformation was used here.
1
vote
1answer
48 views

laplace transform of a sine function

I'm a little confused about how to find Laplace transforms of a sine function when it is a function of time. As in, suppose the function is $x(t)=\sin(at)$ , then I can proceed to get ...
2
votes
2answers
68 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
29 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
1answer
25 views

Proof that laplace's equation is rotationally invariant using chain rule

Suppose $(x, y)$ and $(p, q)$ are coordinates in the plane related by rotation around a fixed point $(a, b)$, as follows: $$\begin{bmatrix} p\\ q\end{bmatrix} = \begin{bmatrix} \cos(t) & -\sin(t) ...
3
votes
2answers
168 views

$y''+2y'+5y=0$, initial value problem with Laplace transform?

here is the question: $$ {\rm y}''\left(t\right) + 2\,{\rm y}'\left(t\right) + 5\,{\rm y}\left(t\right) = 0; \qquad\qquad {\rm y}\left(0\right) = 2\,,\quad {\rm y}'\left(0\right) = -1. $$ ...
0
votes
1answer
26 views

Find Laplace Transform of the following function

How do I find the Laplace transform for the function: $f(t)=t, 0 \leq t \leq 1$ and $2-t, t \geq 1$ I tried looking up the process online, but it remains unclear to me. Thanks in advance!
0
votes
0answers
38 views

s-plane and fourier transform, together in 3d space.

I dont understand how can varying the real part in the s-plane make the amplitude in the fourier plane go to infinity. Lets say the pole is at -3 + -j for example.. Then the laplace transform is the ...
1
vote
2answers
134 views

Laplace Transform of tsin(at) using only the definition

Hello I' am stuck on how to get the final result of the laplace transform of $f(t)=tsin(at)$using (a is a constant) only the definition of $$\int_0^{\infty}f(t)e^{-st}dt$$, I know $sin(at)= {1 \over ...
1
vote
2answers
142 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 ...
1
vote
1answer
33 views

Take the Laplace Transform

Take the Laplace transform of $$ \int_{0}^{t}x^2(x-t)^4 \cos(x)dx .$$ I'm not quite sure where to start...
1
vote
2answers
25 views

Laplace transform using the definition

Find the Laplace of the given function using the definition $$f(t)=tsin(t)$$ I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there ...
2
votes
2answers
47 views

Laplace Transform of an integral

Find the Laplace transform of $$f(t)=t\int_0^{t} \tau e^{-\tau}$$ $L(f)(s)$= ?? My thought is that I can change the $\tau$ to $t$ by Transforming the integral to get $$t/s*L[t*e^{-t}]$$ But ...
1
vote
1answer
39 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
1
vote
2answers
34 views

Find the solution of the IVP using Laplace transforms

The equation is as such: $y''+y=t\sin t$; $y(0)=1, y'(0)=2$ I took the Laplace transform of both sides to yield $F(S)(s^{2}+1)-(s+2)=\frac{-2s}{(s^{2}+1)^{2}}$, and then ...
0
votes
2answers
37 views

Help with basic Laplace Transform - unsure of procedure!!!

I am working on this Laplace Transform, and I've tried looking for a similar example off which to base my own work, but haven't been very successful. I'm confused by the formatting and don't know how ...
0
votes
1answer
33 views

properties of laplace transform

Obtain the transfer function for the following differential equation and check whether the input free solution is stable or not, $$\frac{dx}{dt} + 3x = f(t)$$ Please help, I don't even know where to ...
0
votes
1answer
111 views

Solving $y'-y=2\cos 5t$ using the Laplace Transform

Find the solution to the differential equation, using the Laplace Transform. $y'-y=2\cos 5t$, with initial condition $y(0)=0$. My attempt: First I take the Laplace Transform of each term. ...
0
votes
1answer
28 views

How can we take the LaPlace of a function raised to the power?

For example: $\mathcal{L}$((t-1)^1) Following simple linearity, we achieve the answer. However, following the power of theorem: (I'm not proficient enough in LaTex to write this...) I get the wrong ...
2
votes
2answers
33 views

How can we take the LaPlace transform of a piecewise function?

How can we take the LaPlace transform of a function, given piece-wise function notation? For example, $f(t)=\begin{cases} 0 &\mbox{for } 0<t<2\\ t&\mbox{ for } 2<t \end{cases}$ ...
0
votes
0answers
28 views

How to find the transfer function from the given differential equation

Question: find transfer function from differential eqn $y''(t)+2y'(t)+5=4x(t)$ I am confused about what happens to constant $5$ . will it be zero when we take laplace of whole eqn or not? Can ...