# Tagged Questions

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### Differential Equations with Discontinuous Forcing Functions

$$y''+y'+1.25y = g(t), \quad t > 0,$$ $$y(0) = 0, \quad y'(0) = 0$$ $$g(t) = \left\{ \begin{array}{ll} \sin{t} & 0 \le t < \pi \\ 0 & t \ge \pi \end{array}\right.$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 $2n\pi<t<2(n+1)\pi$ ...
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### 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 ...
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### What Laplace transformation is used for this?

I cannot see the transition here. I was unable to find what Laplace transformation was used here.
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 ...