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

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Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
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1answer
10 views

Finding complete general solution of differential equation with repeated roots (undetermined coefficents)

How do you get a complete general solution for a differential like this? $y^{\prime\prime}+6y^{\prime}+9y=14e^{-3x}$ This is what I have so far for the first part of the problem: $yp=Ce^{-3x}, ...
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1answer
19 views

Exponential of Matirx

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc) to calculate this exponential e^At with A (0 9) (-1 0) I'd ...
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1answer
19 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} \ ...
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1answer
24 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 ...
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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?
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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), ...
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1answer
24 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 ...
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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, $p,c$ ...
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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
21 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 ...
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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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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
21 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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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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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) \\ ...
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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 ...
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1answer
35 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
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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 ...
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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
21 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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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
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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 ...
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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) = ...
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0answers
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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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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 $\mu_y=0$ ...
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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"?
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2answers
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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 ...
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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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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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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 ...
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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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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
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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
47 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? ...
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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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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 ...
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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 $u=\frac{y}{x}\rightarrow y=ux\rightarrow dy=udx+xdu$ $[\csc(u)-u]dx+u(udx+xdu)=0$ ...
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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
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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 ...