In my PDE course, my professor shows me how we can take a generic homogenous polynomial coefficient 2nd order differential equation: $$P(x)y'' + Q(x)y' + R(x)y = 0$$ And transform it into the following expression by diving by $P(x)$ and multiplying by $(x - x_0)$. $$(x-x_0)^2y'' + [\frac{Q(x)}{P(x)}*(x-x_0)](x-x_0)y' + [\frac{R(x)}{P(x)}*(x-x_0)^2]y = 0 $$ Then, we are told that if $x_0$ is a singular point and: $[\frac{Q(x)}{P(x)}*(x-x_0)]$ $[\frac{R(x)}{P(x)}*(x-x_0)^2]$ are analytic at $x_0$, then the singularity at $x_0$ is regular, while otherwise the singularity at $x_0$ is irregular ..read more

Some define $f(t) = (cos(t), sin(t))$ as the solution to the differential equation: $$f''(t) = -f(t)$$ $$f(0) = (1, 0)$$ $$f'(0) = (0, 1)$$ ... or equivalently as the solution to a linear system of first order equations: $$f(t) = (x(t), y(t))$$ $$f'(t) = (-y(t), x(t))$$ $$f(0) = (1, 0)$$ My question is how such definitions get to assume the existence/uniqueness of a solution. Solving with a characteristic polynomial with imaginary roots would assume some pre-existing definition of $e^{ix}$. Using the known Taylor series could demonstrate existence, but then we'd still need non-elementary resul ..read more

I'm reading the very first chapter of Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems from D. V. Anosov, et al. and found myself trying not only to understand, but also to derive the expression presented for "the only possible flow" of $\dot{x} = x^2$ considering that $x(0) = 1$. Well, it's not dificult to solve this IVP, I could express its solution as $x(t) = (1- t)^{-1}$, the maximal interval would be $I(1) = (-\infty, 1)$, of course, but then the text says it's easy to check that the only possible flow is given by the formula $g^{t}x = \frac{x}{1-tx}$. Doe ..read more

$$ \frac{dy}{dx}=\log\left(x\frac{dy}{dx}-y\right)$$ I am trying to solve this equation but was not able to solve. I think this is clairaut's form but can't proceed. Please help ..read more

I've been stuck on this question on James Stewart's "Calculus" (5e Edition) for quite a while. It is under the chapter for differential equations. My current attempt is: I managed to parameterize it, but it's way to messy to continue with... Please send help, thanks ..read more

I recently asked a question related to this one, but different. I am currently struggling with the proof of Lemma 4.2. in this paper (page 16). The context is the following: we are given initial conditions $(x_i(0))_{i \in [n]} \in (\mathbb{S}^{d-1})^n$ ($\mathbb{S}^{d-1}$ is the unit sphere in $\mathbb{R}^d$) such that there exists $w \in \mathbb{S}^{d-1}$ verifying $\forall i, \langle w, x_i(0) \rangle > 0$. The conclusion is that there exists $x^*$ such that $\forall i, \lim\limits_{t\to+\infty} x_i(t) = x^*$ where the functions $x_i$ are solutions of $\forall t, x_i'(t) = \frac{1}{n} \s ..read more

Let $M$ be an orientable compact Rimeannian manifold whithout boundary with volume form $\operatorname{vol}_M$. Take $X\in \mathfrak{X}(M)$ such that if $\phi_t(x)$ is the flow of $X$ and $\phi_1:M\to M$ is such that $$\phi_1^*\operatorname{vol}=\operatorname{vol},$$ does that mean that $\operatorname{div}(X)=0$? I can ask that $\sup_x\lVert X_x\rVert$ to be arbitrarily small if needed. So it seems to me that in this case the divergence should be zero, The fact that this map is volume preserving, we have that its Jacobian is 1. From here we have that this Jacobian is $$e^{\int_0^1\mathrm{div}X ..read more

I was trying to learn the proof of the Existence Theorem for space curves. Can someone give me a specific example of two different curves and tell me how to prove the existence theorem for them curves ..read more

I'm taking a course on ODEs, and I'm currently on the chapter about initial value problems. In this chapter, we try to find out information about the solutions of the problem \begin{cases} x'(t) = f(t, x(t)), \\ x(t_0) = \xi \end{cases} without actually solving it, things like how big the maximal domain of the solutions is, etc. In the particular problem I'm looking for help with, we are given the aforementioned equation with \begin{equation} f(t, x) = \frac{x+t}{t-2} \end{equation} and $t_0 = 0$. We have to make conclusions about the domain of the maximal solutions, when the solutions are inc ..read more

Considering the system $$\frac{dx}{dt} = 2x + x^2 + y^2,$$ $$\frac{dy}{dt} = x^2 - y^2$$ and the region $D = \left\{(x, y) \in \mathbb{R}^2 : x < -1\right\}$, I have to study the existence of periodic orbits in $D$. Trying to apply Bendixson criterion, the divergence of $f(x, y) = \left(2x + x^2 + y^2, x^2 - y^2\right)$ for $(x, y) \in D$, we have that $$\text{div}(f)(x, y) = 2(1 + x - y),$$ so I can't say anything about its sign in $D$. Indeed, I can say there are no periodic orbits in the regions such that $y < x + 1$ or $y > x + 1$. Then what about $D ..read more