Hi everyone,

The exam #3 review sheet is posted under “Classroom Resources/Exam Reviews”. Let me know if you have any questions or corrections.

Regards,

Prof. Reitz

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"...how it differs from the rocks"

Hi everyone,

The exam #3 review sheet is posted under “Classroom Resources/Exam Reviews”. Let me know if you have any questions or corrections.

Regards,

Prof. Reitz

- How to determine if function is differentiable?I am given this piecewise function $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x)= \left\{\begin{array}{ll}x^2 & x>0 \\ 0 & x ≤ 0 \\ \end{array} \right. $ I have to determine if the function is differentiable over $\mathbb{R}$ or not. The way I think about differentiability is like this: "Can a function be differentiated at every point?" But […]
- How to compute $\int\frac1x\Bigl(\sum\limits_{n\ge0}\frac{x^{2n}}{2^{2n}(n!)^2}\Bigr)^{-2}\mathrm dx$?The background of $$\int\frac{1}{x}\left(\displaystyle\sum_{n\geq 0}\frac{x^{2n}}{2^{2n}(n!)^2}\right)^{-2}\text dx$$ is that it yields a second solution to $xy''+y-xy=0$ if you multiply it by the series in the integrand (first solution). My professor found an aproximate answer by multiplying the two series up to $5$th term and then he did long division with $1$ as numerator and the series […]
- What is the best method to fit a function to high dimensional experimental data? Differential equations? Gaussian signal processing? Chaos Attractors? [closed]I apologize for the long post, I just want to provide alot of needed context that might help Background: I am a undergrad conducting waste water treatment research. The idea is looking at varying different process parameters, in order to create a encompassing system dynamics equation. I want to develop a function that relates the […]
- Nonhomogeneous nonlinear differential equation with delta functionsI'm trying to solve the following differential equation $$ y'' + \dfrac{1}{2}(y')^2 = A \delta(x) + B \delta(x-a) + C $$ I tried two times, the first one using Laplace transforms, but I don't really know how to deal with the $(y')^2$ term. I found some papers discussing it, but I couldn't really use them […]
- Finding the analytical solution of a first order system with pure time delayI have a simple system and I am searching for the solution for f(t): $$\frac{\partial f(t)}{\partial t} = c_1 \left( f(t) + g(t) + c_2 \right)$$. It turns out that, in this system $g$ can be related to $f$ by a pure time delay: $$g(t) = f(t-a)u(t-a)$$ where $u(t)$ is the Heaviside function. Applying the […]
- Need help with this differential equation problem involving second derivative and first derivativeI have been trying to solve this problem. I was able to observe that from the given conditions $x^2 + f^2(x) + (f'(x))^2 = c$, where c is some constant, but it does not help me in any of the options. Any hints or ideas would be appreciated Let $f$ be a twice differentiable function […]
- If we perturb an ODE, with the same starting conditions can we show that they converge together over time or at least do not diverge?Suppose we have two ODEs: $\dot{x}(t) = f(x(t),t)$ $\dot{y}(t) = f(y(t),t) + g(y(t),t)$ If we have identical starting conditions $x(0) = y(0)$, we see $$y(t) - x(t) = \int_0^t \left[ f(y(t),t)-f(x(t),t) \right] dt + \int_0^t g(y(t),t)dt $$ Is there analysis giving conditions on $f$ and $g$ that permits us to prove that $\lim_t x(t) - […]
- Relationship between Brownian Motion and Heat Equation?I have heard people say that the Brownian Motion is closely related to the Heat Equation but I am not sure what is meant by this. As I understand, the Brownian Motion is a continuous-time stochastic process such that: Independent Increments: For any sequence of times $(0 \leq t_1 < t_2 < \ldots < t_n)$, […]
- Legendre functions at the end pointsLegendre equation is $$ (1-x^2) y'' - 2xy' + \lambda y=0$$ We are interested in finding solutions in the range $[-1,1]$. We seek solutions around the ordinary point $x=0$ $$ \sum_{n=0}^\infty c_n x^n $$ Since legendre equation has regular singular points at $-1,1$ , this series is guaranteed to converge in $|x|
- $\sum_{n=0}^{\infty} c_{2n}$ and $\sum_{n=0}^{\infty} c_{2n+1}$ given $c_{n+2} = \frac{n(n+1)-\lambda}{(n+2)(n+1)}$Legendre equation is $$ (1-x^2) y'' - 2xy' + \lambda y=0$$ We are interested in finding solutions in the range $[-1,1]$. We seek solutions around the ordinary point $x=0$ $$ \sum_{n=0}^\infty c_n x^n $$ Since legendre equation has regular singular points at $-1,1$ , this series is guaranteed to converge in $|x|

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