Tagged Questions

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

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-2
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0answers
7 views

Differential Equation and dynamical system [on hold]

How to plot the direction field from a autonomous system using mathematica.write command also.
0
votes
0answers
17 views

necessary and sufficient conditions in ODE theory

I have trouble writing proofs when studying the abstract theory of ODE. For instance, I have trouble proving the existence of some special solutions of a given system of nonlinear ODE. In particular, ...
4
votes
1answer
34 views

General solution to $f^{(n)}=f$ but $f^{(k)}\ne f$ for $k<n$

We know that $$\frac{d}{dx}e^x=e^x$$ and $$\frac{d^4}{dx^4}\sin(x)=\sin(x)$$ What is the general solution $f$ to $$\begin{equation} \begin{split} \frac{d^n}{dx^n}f(x)&=f(x) \\ ...
2
votes
1answer
15 views

Growth of plant in greenhouse

The following problem came up in an exam I sat recently. I got 113cm, but I'm quite unsure about my method. Is someone able to go through the working and explain the problem? Of course, I don't ...
1
vote
0answers
15 views

Can anyone help me with this linear differential question?

Given two constant-coefficient operators A and B whose characteristic polynomials have no zeros in common. Let C=AB first part of question is "Prove that every solution of the differential equation ...
0
votes
0answers
14 views

How to check if this solution is equilibrium solution?

Let's have en equation $$ \tag 1 kB(t) = -\alpha \frac{dB(t)}{dt} + \beta \left(\mu (0) - \beta \frac{B^{2}(t) - B^{2}(0)}{2\gamma} \right)B(t), \quad B(0) = b $$ If $\mu (0)\beta = k$ I have an exact ...
0
votes
1answer
11 views

Simple problem about Laplace Equation in a domain

Suppose that "$u$", is solution of the problem $$\triangle u=0, r<R $$ $$u_{r}(R,\phi)=f(\phi), 0<\phi\ < 2 \pi$$ Show that $$\int_{0}^{2 \pi}{f(\phi)d\phi}=0$$ I know what this question ...
3
votes
2answers
17 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
18 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
37 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
22 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
12 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 ...
5
votes
2answers
29 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
20 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
32 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
22 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
33 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
36 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
20 views

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

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
35 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
27 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
25 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
35 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
33 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
27 views

Showing that solution of a differential equation satisfy a relation

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
34 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 $\mathbf{B}$ ...