ordinary differential equations - Intuition or wisdom for stability and instability properties of locally linear system. Boyce, p513, Table 9.3.1

Our instructor requires us to memorize this table for our differential equations exam. So I wonder if anyone has some deeper intuition or observation to help with this? For example, I noticed that whenever eigenvalues $< 0$, the stability (for linear and locally linear systems) is stable. Why is this?@snarski I'm terrified too for the exam. I write about the most general eigenvalue, when $r = \lambda \pm iu$. Solutions to $\mathbf{ x' = Ax }$ are, from BOYCE p161, $x(t) = e^{\lambda t}(\cos(ut) + i\sin(ut)).$ So when $\lambda <0$, then $...Read more

ordinary differential equations - $S(x)=\frac{x^4}{2\cdot 4}+\frac{x^6}{2\cdot 4\cdot 6}+\frac{x^8}{2\cdot4\cdot6\cdot8}+\cdots$

Find the sum of $$S(x)=\frac{x^4}{2\cdot 4}+\frac{x^6}{2\cdot 4\cdot 6}+\frac{x^8}{2\cdot4\cdot6\cdot8}+\cdots$$What I did so far:It's trivial that$$S'(x)-xS(x)=\frac{x^3}{2}-\lim_{n\to\infty}\frac{x^{2n}}{n!2^{{n(n+1)}/2}}$$and $$\lim_{n\to\infty}\frac{x^{2n}}{n!2^{{n(n+1)}/2}}=0$$Solve this ODE I got$$S(x)=\frac{x^4}{8}e^x+Ce^x$$And $S(x)=0$ , So $C=0$.Finally, I got $S(x)=\frac{x^4}{8}e^x$The problem is when I expand the function I got into power series, it looks different from the original one. So I may make some mistakes but I can't tell. ...Read more

ordinary differential equations - For $y''-2\alpha y'+\frac{2}{x} y=0$ , provided that $y(x)=\sum_{n=0}^{\infty} a_nx^n$ show you can determine $a_{n+1}$ from $a_n$

Given the differential equation $y''-2\alpha y'+\frac{2}{x} y=0$, show that if $y(x)=\sum_{n=0}^{\infty} a_nx^n$ is a solution of the differential equation, then $a_0=0$ and that we can determine $a_{n+1}$ from $a_n$ for $n\geq1$So this is the first part of a longer question and I'm already stuck because of the $\frac{2}{x}$. I do know that since $a_0=y(0)=\sum_{n=0}^{\infty}a_n0=0$.The first thing I did was find $y''$ and $y'$ and basically plugged everything in. This gave me the equation$$\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n -\sum_{n=...Read more

ordinary differential equations - Finding a particular solution curve and simplifying

Find the equation of the curve that passes through the point $(1, 3)$ and has a slope of $y/x^2$ at any point $(x, y)$.$$\frac{dy}{dx} = \frac{y}{x^2}$$with the initial condition $y(1) = 3$$$\int \frac{dy}{dx} = \int \frac{y}{x^2} ,~~~y \ne 0,x > 0$$After integrating and solving for $y$, I was able to get$$y = e^{ - (1/x) + C_1}$$but I don't know how the book then gets an answer of$$y = Ce^{\frac{ - 1}{x}}$$I appreciate any assistance in helping me make sense of this simplification....Read more

ordinary differential equations - Help understanding separation of variables technique

Find the general solution of$$(x_{}^2 + 4)\frac{{dy}}{{dx}} = xy$$After separating variables $$\frac{dy}{y} = \frac{x}{x^2 + 4}dx$$What I don't understand is my textbook's result after integrating...$$\int \frac{dy}{y} = \int \frac{x}{x^2 + 4}$$$$\ln \left| y \right| = \frac{1}{2} \ln(x^2 + 4) + C_1$$which they said resulted in...$$\ln \left| y \right| = \ln \sqrt{x^2 + 4} + C_1$$What I want to know is how the $\frac12$ disappeared after integrating and how they came to simplifying it with a square root. I appreciate any constructive insight. ...Read more

ordinary differential equations - Calculating time-varying velocity and final speed through second order ODE

I have to work through an example from physics to strengthen my understanding of Newton's second law of motion through second-order ODEs. I've been provided with the following settings: that the skydiver is descending under a parachute at a steady rate, where t is the time; and I have to assume zero displacement and velocity at time zero. $$\frac{d^2y}{dt^2} + \frac{k}m^{}\frac{dy}{dt} = g$$ I've come to the following general solution y(t): $$y(t) = A + Be^{-\frac{kt}{m}} + g\frac{m}{k}t$$ If I set the displacement and velocity at time zero, I ...Read more

multivariable calculus - Understanding the Test For Exact Differential Proof

Let $M(x,y)$ and $N(x,y)$ be continuous functions with continuous first partial derivatives. Then $M(x,y)dx+N(x,y)dy$ is exact if and only if $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$.I am trying to prove the sufficient direction, assuming $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. The proof says you can either let $\frac{\partial f}{\partial x}=M(x,y)$ or $\frac{\partial f}{\partial y}=N(x,y)$. I will let $\frac{\partial f}{\partial x}=M(x,y)$.Integrating both sides with respect to $x$ gives $f(x,y)=\int...Read more

ordinary differential equations - How to plot a phase space

I was trying to study the function $x(t)=\int_0^t e^{-s^2}ds$, which I did successfully using high school maths. After that, I decided that I wanted to try to study it using some multivariable calculus, but couldn't really do it, so I'm asking for the proper way to face this: the problem is equivalent to the ODE $x'(t)=e^{-t^2}$ given some initial value. Let $y(t)=t$ and therefore $y'(t)=1$. Now I want to study the phase space of the system $\begin{cases} x'=e^{-y^2}\\ y'=1 \end{cases}$. How can I do it? There are no equilibrium points and the ...Read more

ordinary differential equations - Periodic orbit in vector field with positive divergence

The Dulac-Bendixson Theorem states that a vector field with positive divergence defined on a 2-connected set cannot have more than one periodic orbit. I am looking for an exemple of a field defined on an open set minus one point, such that the field has positive divergence everywhere and the differential equation defined by such field has a periodic orbit.Many thanks....Read more

ordinary differential equations - Optimal stopping in red vs black card game deck of 52 cards

I have a optimal stopping problem that is solved by recursion. I was stumped by this question in an interview once. I am hoping someone can walk me through the reasoning so I can reproduce it on similar problems.Imagine you are playing a card game you start with a well shuffled deck of $52$ cards, stacked face down. You have a sequence of turns, $52$ possible turns in total, each turn you either pull the top card and turn it over, or you quit. If you pull a red card you win $1$, and if you pull a blue card you lose $1$. If you played all $52...Read more

ordinary differential equations - Can we derive the PDE followed by a marginal transition probability density?

A pair of correlated stochastic processes follow the SDEs\begin{align}dX_t&=a(t,X_t)\,b(t,Y_t)\,dt+c(t,X_t)\,d(t,Y_t)\,dW_t, &&X_0=\bar{x}\\dY_t&=f(t,Y_t)\,dt+g(t,Y_t)\,dZ_t, &&Y_0=\bar{y}\end{align}where $W_t$ and $Z_t$ are correlated Brownian motions with constant correlation $\rho$ and $a,b,c,d,f,g$ are smooth functions.We know that the probability density $p(t,x,y)$ of the process reaching the state $X_t=x$, $Y_t=y$, given the initial condition at $t=0$ satisfies the forward Kolmogorov equation (also known as Fokker ...Read more

ordinary differential equations - variational problem with constraints

Let me bring to your attention the following problem.Suppose we have the functional $$ F = \int\limits_{a}^{b} f(y(x))\cdot\frac{dy}{dx} dx .$$It is easy to see that that the Euler-Lagrange equation vanishes identically, in other words, every function $y (x)$ is an extremal of the functional. If we write the Euler-Lagrange equation we can see, that it is an identity. Еvery Lagrangian of the form "function multiplied by the derivative of the 1st order" has this property (so called zero Lagrangian). However, the question arises whether the extr...Read more

Limit of y(x) in Second Order Differential Equation

So, even though I know how to solve ODEs, I don't know how I should proceed about this question:Let $a$, $b$, and $c$ positive constants. If $y = y(x)$ is solution to the differential equation $ay'' + by' + cy = 0$, then $\lim_{x\to\infty}$ $y(x)$:(a) doesn't exist and tends to $+\infty$. (b) exists and is $0$. (c) doesn't exist and tends to $-\infty$. (d) exists and is $\pi$. (e) exists and is $e$.I tried to take the limit of the possible solutions but even if $a$, $b$, and $c$ are positive-only numbers, there are many possibilities so I could...Read more