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

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15 views

I want to find Euler-Lagrange equation for the given functional.

I want to find Euler-Lagrange equation for the following: $$J(u) = \int \left( \frac{\psi(x) u + \dot{u}}{\psi(x)u - \dot{u}} \right)dx, \text{where} \ \psi(x) \ \text{is an explicit function of} \ ...
1
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1answer
17 views

calculate the second derivative using `ode45`

I have a second order differential equation. I am using ode45 to solve the problem. ode45 converts the equations to the first ...
1
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1answer
27 views

why are these two power series the same

$$-\sum_{\color{red}{n=1}}^{\infty}nc_{n}x^{n}=-\sum_{\color{red}{n=0}}^{\infty}nc_{n}x^{n}$$ How come one starts at $1$ and the other starts at $0$ yet their equal? Do they both equal infinity?
3
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0answers
14 views

Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(t,x)-2Y_{tx}(t,x)+Y_{tt}(t,x)=0, \quad 0<x<1, t>0$$ $$Y(0,x)=Y_t(0,x)=0, \quad 0<x<1$$ $$Y(t,0)=0, \ Y(t,1)=F(t), ...
0
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1answer
23 views

What type of differential equation is that?

Good day. I can't understand what type this DE has $ (7x-8y)y'=2x^2-y $ I guess it can't be homogeneous or separable equation. And it seems what it is not a linear equation.. Maybe it is Bernoulli ...
3
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1answer
20 views

Solving a system of Differential Equations: arbitrary constants

For a research project I am carrying out I am required to solve the system: $\frac{dp}{dt} = -lp $, $ \frac{dc}{dt} = lp - kc $ with initial conditions $p(0) = p_0 $ and $c(0) = 0 $. Here, $p,c$ ...
4
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1answer
27 views

Existence and uniqueness of soluctions of $y'=xy^{2/3}$

It is asked to analyze the existance and uniqueness of solutions of the ode at every point $(x_o, y_o)$ $$y' = 3y^{2/3}$$ My attempt: We consider the initial condition $ y(x_o)=y_o$. If we consider ...
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1answer
20 views

Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
0
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1answer
10 views

Differential equations resonance

I've got the question 'Solve (c^2)y ' ' + y = 0, y(1)=1, y'(0)=0. Show that as c->0, the solution does not tend to a limit'. From solving the equation I got the roots as +-(1/c)i, and then using set ...
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2answers
28 views

When do you drop the absolute value from ln|x| + C when integrating $\frac{1}{u}du$

Given: p(t) represents the number of cats, when t>=0. Given: p(t) is increasing at a rate directly proportional to $800-p(t)$ So, I represent this as: $\frac{dp}{dt}= k(800-P)$ I want p(t), so I ...
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0answers
21 views

Asymptotic behavior of the solution of a 2nd order linear ordinary differential equation

In studying the harmonic oscillator, we encounter the equation $$ f'' +(2E - x^2) f = 0$$ What is the asymptotic behaviour of the solution to this equation for a generic $E$? Any good book on ...
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1answer
24 views

solvability condition for differential operator

While reading the research article I came across following derivation, given a self-adjoint operator, \begin{eqnarray} L = \frac{d^2}{dx^2} + f(x) \end{eqnarray} \begin{eqnarray} L\psi_1(x) ...
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1answer
20 views

Uniqueness of differential equation solutions

I need to solve this DE $$y'' - 2x^{-1}y' + 2x^{-2}y = x \sin x \tag{*}$$ I found the complementary functions to be $x^2$ and $x$, and also noticed by guessing that the particular integral is $y = - ...
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0answers
12 views

Differential Equation and dynamical system [on hold]

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

necessary and sufficient conditions in ODE theory [on hold]

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
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1answer
41 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
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1answer
20 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 ...
2
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1answer
34 views

If a differential operator $C$ factors as $AB$, then every solution of $C(y)=0$ has the form $y=y_1+y_2$ with $A(y_1)=0$ and $B(y_2)=0$

Given two constant-coefficient operators $A$ and $B$ whose characteristic polynomials have no zeros in common. Let $C = A B$. First part of question is Prove that every solution of the ...
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1answer
14 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
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2answers
24 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 ...
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1answer
20 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 ...
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1answer
39 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} ...
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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
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2answers
43 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 ...
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0answers
14 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
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2answers
31 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
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0answers
16 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 ...
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0answers
23 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
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1answer
40 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
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2answers
28 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
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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 ...
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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 ...
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0answers
19 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
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1answer
36 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 ...
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0answers
17 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 ...
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2answers
26 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 $$y'=c_1e^{-\frac{2}{3}x^{\frac{3}{2}}}$$
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1answer
34 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 $dy$? ...
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2answers
46 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'= ...
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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
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1answer
31 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 ...
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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.
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2answers
36 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
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2answers
29 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
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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 ...
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0answers
18 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?
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1answer
28 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 ...
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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
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1answer
26 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
26 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: ...