I have a conflicting commitment in the Math Department. However, I should be able to be in class a few minutes early if you have questions.
See you in a bit,
Prof. Reitz
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- Compute $\int_0^\infty f(x)dx$ where $4f''(x) + 4f'(x) + f(x) = \frac{1}{e^x - 1}$ and $\lim_{x\rightarrow \infty}e^{\frac{x}{2}} f(x) =0$.[Cambridge University Integration Bee 2023 - Relay Round] Suppose $4f''(x) + 4f'(x) + f(x) = \frac{1}{e^x - 1}$, with $e^{\frac{x}{2}} f(x) \to 0$ as $x \to \infty$. Evaluate $\int_0^\infty f(x) dx$. I tried to actually use differential equations, but WolframAlpha thankfully warned me this has special functions if I ever try to find the actual […]
- Solving the Submarine-Destroyer Problem using only geometryGeorge Simmons, in his classic book on differential equations, poses this problem: A destroyer is hunting a submarine in a dense fog. The fog lifts for a moment, discloses the submarine on the surface 3 miles away, and immediately descends. The speed of the destroyer is twice that of the submarine, and it is known […]
- First order non-linear differential equations [closed]I was trying to solve this problem whole night, trying to replace $\frac{dy}{dx}$ with $p$, but could not find any success. So, here's my equation $$\frac{dy}{dx} = \frac{\sqrt{2x^2 + y^2 + y}}{x}$$ I will be grateful for any help
- Is the Frobenius power series solution valid for complex variable ($x=e^{i\;\theta}$ ) too?Are we allowed to use the Frobenius power series solution for an ODE $$ A(x) \;\Phi''(x)+B(x) \;\Phi'(x)+C(x)\; \Phi(x)=0$$ in which $x=e^{i\;\theta}$ is a complex unit circle? In general, my question is: is the Frobenius power series solution valid for complex variables too?
- How does Stokes First Problem Yield the Error Function in its Solution?When solving Stokes first problem using a similarity variable, I get to the point where I have converted the PDE (from Navier-Stokes Eqn) into an ODE, and when I go to solve it the solution does not look like the error function. The ODE I get using $\eta$ as a similarity variable looks like: $f'' […]
- General solution of $P(s) T = 1$Suppose that we are given the distribution equation $$P(s) T = 1, \tag{1}$$ where $P(s)$ is a polynomial in $s$ and $T$ is a distribution. What's the solution for $T$ in this equation? I want to find a rigorous answer, rather than a heuristic or intuitive answer. Actually this equation comes up when we try […]
- Is this the correct Jacobian matrix for the following system?As I'm working on a homework problem, I have the following system \begin{cases} \dot{x}=y\\\dot{y}=-x-\epsilon(\alpha x^2\operatorname{sgn}(x)+\beta x^3) \end{cases}In order to find the Jacobian matrix, I've got $$J=\begin{bmatrix} \frac{\partial x'}{\partial x} \frac{\partial x'}{\partial y} \\ \frac{\partial y'}{\partial x} \frac{\partial y'}{\partial y} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$ since this is a linear system, I've […]
- Since linear ODEs have a general solution, when are integrating factors needed?Linear homogenous ODEs with constant coefficients have straightforward general solutions; common classes of inhomogeneous have general solutions as well. Integrating factors are only useful for linear ODEs. When do we need them? Why can't we just use the general solution of linear ODEs? Are integrating factors only needed when the ODE does not have constant […]
- Does the control function impact linearization about a point in a dynamics system?Let's say I have a standard dynamics system for position ($x$)/velocity ($x'$)/acceleration ($x''$) along with a controller torque $u$. How would I determine if a particular dynamics equation is linearizable about a particular point if the impact of the control function is different? I have the following three cases and I am looking to see […]
- $C^k$-Norm Bounds for the Flow of an ODEI have a question about the regularity of the flow map associated with a time-dependent ODE. Let $v \in C^k(\mathbb{R}^d \times \mathbb{R}, \mathbb{R}^d)$ be a time-dependent vector field mapping $(x,t) \mapsto v_t(x)$, which is continuous in $t$ and Lipschitz continuous in $x$. Consider the ODE: \begin{cases} \frac{d}{dt} x(t) = v_t(x(t)), \\ x(s) = x_s. \end{cases} […]
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