**WeBWorK:** Assignments `14-InverseLaplaceTransforms,`

and `16-LaplaceIVP`

are due Tuesday, 5/24, at midnight.

**OpenLab:** OpenLab #3, Advice for the Future, is due on Tuesday, 5/21 (the day of the final exam).

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"...how it differs from the rocks"

**WeBWorK:** Assignments `14-InverseLaplaceTransforms,`

and `16-LaplaceIVP`

are due Tuesday, 5/24, at midnight.

**OpenLab:** OpenLab #3, Advice for the Future, is due on Tuesday, 5/21 (the day of the final exam).

- On the maximum rate of change of solutions to the wave equation, Could it be limited?On the maximum rate of change of solutions to the wave equation, Could it be limited? It is highly probable I have a huge conceptual mistake: I am going to redact the question as I believe things works in order to formulate the question, so please be patient, I believe the core is going to […]
- The logic subtlety behind solving differential equations.Let me first explain what has led me to ask this question. When solving functional equations, it is often the case that through a link of implications (that is, uni-directional implications), we get several possible solutions for the functional equation. Then, we have to plug these functions into the original equation to see whether each […]
- An autonomous ODE solution has a 0 derivative for some t. Is it an equilibrium solution?Given an autonomous ordinary differential equation, suppose we know a solution X(t) has 0 derivative at some value of t. Does this mean X is necessarily an equilibrium solution? Drawing slope fields for autonomous ODEs, it seems to me to the the case. A 0 derivative means an horizontal line composed of horizontal arrows. So […]
- Solution to ODE doesn't blow up to infinityThe problem is Show that for any $a \gt 0$, The boundary value problem $y'' = y^2, y(-1)=a, y(1)=a$ has a solution. To prove this I defined y_b to be the unique solution to the ODE $y(0)=b, y'(0)=0, y'' =y^2$ If $b = 0$ then we have that $y_b = 0$ and so $y_0(1)
- Existence and uniqueness of a limit of solutions of integral equationsLet $X_t$ be a solution to the equation \begin{equation} \label{eq1} (*) X_t=x_0+\lim_{n \to \infty} \int_0^t f_n(X_s,s) ds, \end{equation} where $x_0 \in \mathbb{R}$ and $f_n(X_s,s)$ is integrable for each $n$, not necessarily continuous, and where $\lim_{n \to \infty} f_n(X_s,s)$ does not converge to an integrable function. Furthermore, for each $n$, there exists a unique continuous solution […]
- When does a Koopman Operator exist for a Dynamical System?Koopman Operators have been popular approach for studying and learning dynamical systems for the past five years or so, especially in a data-driven landscape. However, I can't help but feel like it is being advertised as a silver bullet to tackle any dynamical system. My intuition tells me there must be some Dynamical Systems where […]
- Differential Equations Describing Orbits of Particle Motions [closed]I am an mechanical engineer. In my study of particle dynamics, I got the some patterns of particle motion ( the link to the image is given below ): The first shows the orbit of three sine angles, and the second shows the orbit of three angular velocities in body-fitted frame, and the third shows […]
- Why does the solution to $x'=x^3$ grow less fast than the solution to $x'=x^2$?Reading through Schaeffer and Cains book on ODE's I made up the following example to test my understanding of a passage (see below). Consider two IVP's: $x'=x^2$ and $y' =y^3$. Then by seperability $-1/x = t+C \iff x= \frac{1}{-C-t}$. Likewise $y = \pm \sqrt{\frac{1}{2(-C-t)}}$. For our example consider C=-1. Thus $x= \frac{1}{1-t}$ and $y=\sqrt{\frac{1}{2-2t}}=\sqrt{2}\sqrt{\frac{1}{1-t}}/2$. On […]
- Differential equation with Zigzag numbers.I'm given the recurrence relation $$ 2E_{n+1} = \sum_{j=0}^{n} \begin{pmatrix} n \\ j \end{pmatrix} E_j E_{n-j}, \quad n = 1, 2, \dots $$ with $E_0 = 1$ and $E_1 = 1$. And the function $$ F(x) = \sum_{n=0}^{\infty} \frac{E_n}{n!}x^n $$ I am also given the identities $$ \left( \sum_{n=0} a_n x^n \right) \left( \sum_{n=0} b_n […]
- Why can we input initial conditions into the CF before finding the PI when using Green's Function?For any given ODE of the form $x_{p}''(t)+q(t)x_{p}'+r(t)x_{p}(t)=f(t)\quad\forall t>t_{o} \quad \text{subject to } \text{$\begin{cases} \,x(t_{0})=0\\ x'(t_{0})=0 \end{cases}$}$ We may break the solution into two parts: $x(t)=x_c(t)+x_p(t)$ Where $x_c$ is the complementary function and $x_p(t)$ is the particular solution aka the particular integral. Usually for IVPs we must first find $x_p(t)$ and apply the initial conditions […]

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