Assignments Week 3

February 14, 2019 / Jonas Reitz / 0 Comments

WeBWorK: Assignment 5-Exact Due Tuesday 2/19 at midnight
NOTE: As announced in class today, the WeBWorK assignments from last week have been extended until next Tuesday, 2/19 at midnight:
3-Separable , 4-Bernoulli-nosub and 4b-Homogeneous-YoverX due Tuesday 2/19 at midnight.

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assignments week 3

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Question on mass-spring system
Q. Suppose there is no damping in a mass and spring system with $m=5$, $k=20$ and $F_0=5$. Assume $w$ is chosen to be precisely the resonance frequency. Find the amplitude of the oscillations at time t=100, assuming the system is at rest at t=0. The differential equation for this question is $5x''+20x=5cos(2t)$. Upon solving it […]
Sign for SHM velocity squared function of x, given the initial conditions. Book error?
I am having an issue. the problem after I solve this differential equation, using vdv/dx = a then I obtain a solution for v: $v=±{\sqrt (144-4x^2)}$ Then because of the initial condition that x=0 (Its in SHM so the centre can be easily deduced as o using the SHM definition of acceleration) and v=12, I […]
Find Critical Points in differential equation [closed]
$\left.\begin{array}{rcl} {\displaystyle\operatorname{y}'\left(t\right) -qt\operatorname{y}\left(t\right)} & {\displaystyle =} & {\displaystyle 1} \\ {\displaystyle\operatorname{y}\left(0\right)} & {\displaystyle =} & {\displaystyle y_{0}} \end{array}\right\}\quad$ Solve this equation and assuming that $y_{0}$ is near $-1$.
Can you solve this recursive differential equation? [closed]
Equation Sorry for not giving a lot of information about the differential equation because i dont have any :)
ODE book for a computer science researcher (Birkoff/Rota vs Arnold)
I've have been looking for a book on ODEs and have narrowed it down to 2 candidates: Birkoff and Rota's 'ordinary differential equations' or Arnold's 'ordinary differential equations: 3rd edition'. From reading online and skimming the texts, it is my understanding that B&R's book teaches a traditional course on ODEs, full of all the tricks […]
Could someone explain the reason behind using Legendre Polynomials?
Where $x = \cos(\phi)$ and therefore (and this is very important) $x=[-1,1]$, we have the (special such that m = 0) associated Legendre Equation $(1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx}+ky = 0$, and the power series solution obtained via the recurrence relation $c_{n+2} = \frac{[n(n+1)-k]}{(n+2)(n+1)}$c_n is: $$y(x) = c_0\bigg[1-\frac{k}{2!}x^2-\frac{k(6-k)}{4!}x^4+...\bigg] + c_1\bigg[x+\frac{2-k}{3!}+\frac{(2-k)(12-k)}{5!}x^5+...\bigg]$$ The problem with the series […]
Proving an inner product equality related to Ito process
I'm taking a course in stochastic differential equations and I'm having a trouble proving a fact that was stated as a middle step in discussing Ito processes without a proof in one of the lectures. Namely, in a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ and for $W_s$ a Wiener process and $\Theta, \Gamma$ adapted processes the […]
How to solve a spring-mass differential equation with impulse at $t=\pi$ without Laplace transform?
Consider the differential equation $$x''+x=F_0\delta(t-\pi)$$ with $x(0)=0$ and $x'(0)=1$. I can solve this using Laplace transform. The result I find is $$x(t)=\begin{cases}\sin{(t)}\ \ \ \ \ &t\leq \pi \\ \sin{(t)}(1-F_0)\ \ \ \ \ &t\geq \pi\end{cases}$$ My question is about how to solve this differential equation without Laplace transform. Let's think about the system as […]
Given a function, find a Stochastic Differential Equation that the function satisfies
Suppose we have an SDE: $$dX = A X dW_1 + B X dW_2, \tag{1}$$ where $A$ and $B$ are constant non-commuting matrices acting on the Hilbert space $\mathbb{C}^2$, and $W_1(t), W_2(t)$ are independent 1-dimensional Wiener processes. Also, assume the initial condition is a constant vector $X_0 \in \mathbb{C}^2$. Since $A$ and $B$ do not […]
Neural network that learns ODE's 'refuses to learn' initial conditions [closed]
I have implemented a simple network that for now i'm just trying to teach the ODE: $$\frac{\text{d}x}{\text{d}t} = x$$ Using the simple code below: import tensorflow as tf from tensorflow.keras.models import Sequential from tensorflow.keras.layers import Dense from tensorflow.keras.optimizers import Nadam import numpy as np #Simple NN model = Sequential([ Dense(5, input_dim=1, activation='relu'), Dense(5, activation='relu'), Dense(1) […]

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Assignments Week 3

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