Tagged Questions

Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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3
votes
1answer
10 views

Show stable node or spiral cannot occur

If I have the equation: $$\ddot{x} + f(\dot{x}) + g(x) = 0$$ where $f$ is even and $f$ and $g$ are both smooth, how do I show that the equilibrium points cannot be stable nodes or spirals? What I've ...
0
votes
1answer
14 views

Systems of First Order Linear Equations, finding P(t) from two given vectors

Consider the vectors $x^{(1)}(t) = (t,1)$ and $x^{(2)}(t) = (t^2, 2t)$ I computed the Wronskian which is t^2. I also know that it's continuous everywhere except when t=0. But I was wondering how to ...
1
vote
1answer
35 views

How do first integrals help you solve differential equations?

I am reading about Euler-Lagrange equations and this particular section is a little unclear. Consider the differential equation $$\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix} = \begin{bmatrix} ...
-3
votes
0answers
20 views

Find the eigenvalue and eigenfunction of the boundary value problem

By setting $y=\frac{u}{\sqrt{x}}$, find the Eigenvalues and Eigenfunction for a boundary value problem: $$y'' + \frac{y'}{x} +\Big(λ- \frac{1}{4x^2}\Big)y = 0 ,\ \ y(\pi)=y(2\pi)=0$$ The only ...
3
votes
2answers
42 views

Why is $f(x) = x + \frac{1}{x}$ a mapping contraction?

Why is $f(x) = x + \frac{1}{x}$ a mapping contraction? The metric space in question is $[1,\infty)$. Also, if this were a contraction, wouldn't it have a fixed point by Banach's theorem? It looks to ...
1
vote
0answers
11 views

Differential equation and integration approximation magic

Say we have a differential equation: $$df(\mu) = g(f(\mu))dv(\mu)$$ I was wondering under what conditions we get something like this (integrating from $\mu_1$ to $\mu_2$): $$\int df(\mu) \approx ...
4
votes
0answers
19 views

Differential Equations Constant

The function $y(x)$ satisfies the linear equation $$y'' + p(x)y' + q(x)y = 0.$$ The Wronskian $W(x)$ of two independent solutions, denoted $y_1(x)$ and $y_2(x)$, is defined to be $$W(x) = ...
0
votes
0answers
11 views

How to reduce order of this ODE

I want to reduce this ODE to lower order but I am confused in some steps. Can someone comment? $$ AB\frac{d^3u}{dz^3}+C(D-z)\frac{du}{dz}=0, \,\, 0<z<L $$ $A,B,C,D,L$ are constants, all ...
1
vote
0answers
18 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
0
votes
1answer
39 views

What does “two polynomials have no zeros in common” mean?

The question is Given two constant-coefficient operators $A$ and $B$ whose characteristic polynomials have no zeros in common. Let $C=AB$... What does that mean by "no zeros in common"?
4
votes
2answers
27 views

Ordinary differential equations of the form $M(x,y)dx+N(x,y)dy=0$ question

An ODE of the form $M(x,y)dx+N(x,y)dy=0$ is called "good" if $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$ We are given the differential equation ...
0
votes
1answer
15 views

If $F(t,x)$ decreases in $x$ for every $t$, show that if $f,g$ satisfy the equation $x' = F(t,x)$, then $|f(t)-g(t)|$ monotonically decreases.

Given a decreasing function $F(t,x)$ by $x$ for every $t$, show that if $f,g$ satisfy the equation $x' = F(t,x)$, $|f(t)-g(t)|$ monotonically decreases. I've tried deriving, I've tried plugging in ...
1
vote
0answers
11 views

countable zeros of a particular solution to some 2nd order differential equation

Consider the differential equation$: \ e^xx^2y''-e^xxy'+(x^2-1)y=0.$ Suppose $f:(-\infty,0) \to \mathbb{R}$ is such that $(1-x^2)f(x)=e^x(x^2f''(x)-xf'(x)), \forall x\in (-\infty,0).$ If $f$ is not ...
1
vote
0answers
17 views

Find the leading order uniform approximation to the boundary value problem $\epsilon y''+y'\sin x+y\sin 2x = 0$? [duplicate]

$$\epsilon y''+y'\sin x+y\sin 2x = 0$$ with boundary conditions $y(0)=\pi$ and $y(\pi)=0$ as $\epsilon \rightarrow 0$. I don't know how to find out where the boundary layer is? I thought initially it ...
0
votes
1answer
31 views

How do I solve this system of differential equations? $\frac{dy}{dx}=\frac{-y}{x}+x z, \frac{dz}{dx}=\frac{-2y}{x^3}+\frac{z}{x}$ [on hold]

How do I solve this system of differential equations? $$\left\{\begin{align}\frac{dy}{dx}&=\frac{-y}{x}+x z,\\ \frac{dz}{dx}&=\frac{-2y}{x^3}+\frac{z}{x}\end{align}\right.$$ So, I have quite ...
0
votes
0answers
14 views

How to prove this property?

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
0
votes
2answers
21 views

How do you solve: $y'=c_1e^{-\frac{2}{3}x^{\frac{3}{2}}}$?

How do you solve that ODE? I understand it has gamma functions but I have no clue where to start. Thanks $$y'=c_1e^{-\frac{2}{3}x^{\frac{3}{2}}}$$
1
vote
1answer
31 views

$dx$-notation in analysis

In the context of integrals and differential equations, often the symbol $df$ or $dy$ appears, where in some previous steps $f$ and $y$ were functions. What do these symbols mean $df$ and $dy$? ...
3
votes
2answers
34 views

Find $a,b$ to make $V$ a Lyapunov function

Given $V(x,y)=ax^2+by^2$ I'm asked to find $a$ and $b$ to make $V$ a Lyapunov function for the following systems: $(1)$\begin{cases} x'= -x^\color{red}{3}+xy^2 \\ y'= ...
0
votes
1answer
28 views

First order ODE: $y'=\frac{b\sqrt{x^2+y^2}-ay}{ax}$

I need to solve the ODE $$y'=\frac{b\sqrt{x^2+y^2}-ay}{ax}$$ I've tried the substitution $ y = x u(x) = ux$ but, even ignoring the modulus of x, I was't able to solve it. Any other suggestion? ...
0
votes
1answer
30 views

Two Body Orbit Problem [on hold]

I really need help urgently. What I've got are two different circles with their radius coming from a fixed center point. The two radius's which can be considered as a line are being rotated at a ...
-1
votes
0answers
16 views

How to prove the operator D=d^(4)/dx is self adjoint

I'm trying to prove $D=d^{4}/dx$ is self adjoint, I think it is trivial but the book let me use Lagrange identity to show it.
1
vote
2answers
30 views

Combinations of fruits and their “nutrients”

As a computer scientist and not a mathematician, I know not some of the formal language to describe my problem, so I'll present it in a word problem form. Maybe someone can help me hone my search and ...
0
votes
2answers
28 views

Solve this Differential Equation $[x\csc(\frac{y}{x})-y]dx+ydy=0$.

$[x\csc(\frac{y}{x})-y]dx+ydy=0$ My work: $[\csc(\frac{y}{x})-\frac{y}{x}]dx+\frac{y}{x}dy=0$ Let $u=\frac{y}{x}\rightarrow y=ux\rightarrow dy=udx+xdu$ $[\csc(u)-u]dx+u(udx+xdu)=0$ ...
2
votes
0answers
39 views

Annoying differential equation involving composition

Upon trying to crack into a problem, I managed to end up with the following differential equation. $$ y = xy' - y'\circ y', \qquad\text{or}\qquad y(x) = x\cdot y'(x) - y'(y'(x)) $$ I haven't a clue ...
-2
votes
0answers
17 views

What is symmetric differential equation? [on hold]

What is the meaning of Z2-symmetric differential equation? and genericaly What's the meaning of symmetry about differential equation?
0
votes
1answer
26 views

How to determine $2\pi$ periodic function?

Let $f(t) = 2\pi \sin t$, and determine a $2\pi$-periodic function $y^∗$ with the property that $\lim_{t\to+\infty} |y(t) − y^∗(t)| = 0$ for every solution $y$ of $y′ + y = f$. I am having trouble ...
2
votes
1answer
31 views

Only isolated critical points can be asintotically stable.

For an equation of the form $\dot{x}=f(x)$ I'm asked to prove that is not possible for a not isolated critical point $a$ be asintotically stable. Is this statement wrong? Because what it asks not only ...
-3
votes
0answers
25 views

Boundary Value Problems for the Heat Equation [on hold]

This tasks are from the book Jeffery Cooper, Introduction to Partial Differential Equations with MATLAB.
3
votes
1answer
24 views

How do you solve a 2nd order differential equation of the form $v = v' - v'' +C^t +D^{t+E}$

I've been working on an economic simulator for a game I've been making and in order to simulate the velocity of money, I created the differential equation of the form $v = v' -v'' + C^t + D^{t+E}$. ...
5
votes
1answer
23 views

Fourier Transform of Newton's Law of Cooling

I am attempting to solve Newton's Law of Cooling differential equation with Fourier Transforms for a high school math report. Can Fourier Transforms be used to solve first-order ODEs? The equation is: ...
4
votes
1answer
34 views

What are the equations modelling a vertical spring system with two masses?

Modeling a vertical spring system with one mass is a pretty common problem. I looked around online and found some horizontal spring systems with two masses, but no examples of a vertical one. I'm ...
1
vote
2answers
30 views

Characteristics method applied to the PDE $u_x^2 + u_y^2=u$

I am trying to solve: $u_x^2 + u_y^2=u$ with boundary conditions: $u(x,0)=x^2$. Unfortunately it leads to equations that makes no sense (sum of squares is $0$ and all constants are $0$). I would be ...
0
votes
0answers
32 views

Systems of First Order Linear Equations - Differential Equations

Consider the vectors $x^{(1)}(t) = (t,1)$ and $x^{(2)}(t) = (t^2, 2t)$ I computed the Wronskian which is t^2. But I was wondering how to solve the following questions: 1) In what intervals are ...
0
votes
0answers
17 views

I am required to solve the boundary value problem $y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$ using the midpoint method.

I am required to solve the boundary value problem $$y'' = 4x^2y' + 2xy,\space y(1) = 4,\space y(2) = 2$$ using the midpoint method. In order to get two first order equations I have set $u_1=y\space ...
0
votes
2answers
39 views

Help with Runge-Kutta method for solving systems of differential equations

I am currently doing an investigation with SIR model for predicting the progress of an infectious disease. However, I am not very much familiar with systems of differential equations,so I would need ...
2
votes
4answers
50 views

Solving $y'(x) = \frac{y(x)}{3x-y^2(x)}$?

Solving $y'(x) = \frac{y(x)}{3x-y^2(x)}$ ? I'm trying to solve this first order non-linear equation. I've tried to plug in a couple of different things and would appreciate if anyone could point me in ...
2
votes
1answer
32 views

''Differential equation'' with known solution $\sin$ and $\cos$

I am given the following two two equations $f,g : \mathbb{R} \to \mathbb{R}$ are differentiable on $\mathbb{R}$ and they satisfy $\forall x,y \in \mathbb{R}$ $$f(x+y) = ...
1
vote
0answers
23 views

Showing $\lim_{x\to\infty} [(\phi(t)-sin(wt))^2+(\phi'(t)-wcos(wt))^2]=0$

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^1 |b(t)| dx <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{x\to\infty} ...
1
vote
0answers
26 views

Stability of $a$ implies $\lim _{t\to \infty} x(t)= a$

I have the differential equation $x'=f(x),x\in\mathbb{R}^n$. Let $a$ be a stable point of the differential equation, I want to prove that if $x(t)$ is a solution such that $\forall ...
3
votes
2answers
32 views

EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
-2
votes
0answers
15 views
1
vote
0answers
33 views

Initial Value for an ODE Problem

I have the following ODE $\mathbf{A}\dfrac{d\vec{y}}{dt}+\mathbf{B}\vec{y}=\vec{x}$, where $\vec{y}$ and $\vec{x}$ are $n\times1$ vectors and are functions of $t$, and $\mathbf{A}$ and $\mathbf{B}$ ...
2
votes
1answer
19 views

Fourier Series Coefficient

I am trying to review the basics. Find the Fourier series for the function $$f(x) =\left\{ \begin{array}{l l} 2x & \quad -\frac{\pi}{2}<x<\frac{\pi}{2}\\ 0 & \quad ...
1
vote
0answers
22 views

Solving a homogeneous linear system of differential equations: no complex eigenvectors?

I have to solve the following equation by diagonalization. $ X' = \begin{bmatrix}1 & 1\\1 & -1\end{bmatrix} X$ I was able to determine the complex eigenvalue roots: $det(A-\lambda I)=0$ ...
3
votes
1answer
14 views

How does the recursion relation work in the solution to this differential equation (using series)?

Sorry for the vague title but it would not let me post the first step and last step of this equation (too many characters!). How does $$\dfrac{a_0}{3n(3n-1)(3n-3)(3n-4)\cdots 9 \cdot 8 \cdot 6 \cdot ...
1
vote
0answers
16 views

Nonhomogenous differential equations

I have 2 nonhomogenous differential equations ($\alpha,\beta, c, d$ are constants and $z,y,z_2,y_2$ are functions of x) 1) $z_2'(x)-\alpha[z_2(x)-2z_2(x+c)-y_2(x+c)]=-2\alpha z(x+c)[2y(x+c)+z(x+c)]$ ...
0
votes
2answers
27 views

Does Runge Kutta need future state of system?

In order to use the RK methods, you need to know the state of the system at future time-steps which can be expensive to compute (e.g., in physics simulations). As a simple example I'll use RK-2: In ...
0
votes
3answers
28 views

IVP with Laplace Transform

My attempt: Y = Laplace $$s^2Y -sy(0) - y'(0) - 3Y = ??$$ How do I set up $$h(t)$$ in the form of laplace?
-3
votes
1answer
53 views

Show that $y'=x|y|$ has unique solution

Show that the ODE $$y'=x|y|$$ has an unique solution for all $ (x,y) \in \mathbb R^2$. My attempt: I am not sure if I should consider the two subsets of the domain that sepair $y\leq 0$ and $y ...