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

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6
votes
2answers
560 views

Legendre polynomials, Laguerre polynomials: Basic concept

I am asking a simple conceptual question. I saw in many Mathematics and Mathematical physics text books that the Legendre polynomials and Laguerre polynomials "falling from the sky"! I mean, I didn't ...
6
votes
2answers
4k views

Can a differential equation have non unique solutions?

There are theorems of existence and uniqueness of differential equations. I was wondering if it is possible that a differential equations has a solution but it is not unique.
5
votes
1answer
138 views

Solutions of autonomous ODEs are monotonic

Problem. Let $I,J$ be open intervals, $\,f:I\to \mathbb R$, continuous, $\,\varphi :J\to \mathbb R$, continuously differentiable, with $\varphi[J]\subset I$, and $\varphi$ satisfying $$ ...
5
votes
2answers
261 views

How to prove $(x^2-1) \frac{d}{dx}(x \frac{dE(x)}{dx})=xE(x)$

$$E(x)=\int_0^{\frac{\pi}{2}} \sqrt{1-x^2 \sin^2 t}\, dt$$ Where $E(x)$ is complete elliptic integral of the second kind. $u=\sin t$ $$E(x)=\int_0^{1} \frac{\sqrt{1-x^2 u^2}}{\sqrt{1-u^2}}\, du$$ ...
5
votes
2answers
567 views

Regarding Ladder Operators and Quantum Harmonic Oscillators

When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator: Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
4
votes
1answer
68 views

Find minimizer of the functional $l(u)= \int_{-1} ^1 u(t) \mathbb d t$

Find minimizer of the functional $ l(u)= \int \limits _{-1} ^1 u(t) \mathbb d t $ with $u(-1)=u(1)=0 $ subject to $g(u)=\int \limits _{-1} ^1 \sqrt{1+u'(t)} \mathbb d t=π $. I solved it using ...
4
votes
1answer
205 views

Solution of $ f \circ f=f'$

Let $f:\mathbb R \to \mathbb R $ be a function such that $f \circ f=f'$ and $f(0)=0$ , I proved that $f$ is the null function. Can we prove that the same result holds if we change $f \circ f=f'$ by ...
4
votes
3answers
255 views

Prove that if $\phi'(x) = \phi(x)$ and $\phi(0)=0$, then $\phi(x)\equiv 0$. Use this to prove the identity $e^{a+b} = e^a e^b$.

I am given the following. hint Consider $f(x)=e^{-x} \phi(x)$. I am unsure how to approach this problem.
4
votes
1answer
6k views

Series solution to $y''-xy'-y=0$

So I'm learning to solve ODE's with series on my own using Boyce and DiPrima and exercise #3 is irking me...just looking for power series solutions around the ordinary point... $$y''-xy'-y=0$$ So I ...
4
votes
1answer
201 views

How do you solve this differential equation using variation of parameters?

$\color{green}{question}$: How do you solve this differential equation using variation of parameters? $$y"-\frac{2x}{x^2+1}y'+\frac{2}{x^2+1}y=6(x^2+1)$$ $\color{green}{I~tried}$ . . . ...
4
votes
1answer
386 views

Sturm-Liouville Questions

In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on. The first issue I have is that my book began with the statement that given $$L[y] = a(x)y'' + b(x)y' + ...
3
votes
1answer
43 views

Boundary value problem and twice differentiable solutions

Let's consider a boundary value problem $$ u''(x) = f(u(x)) + g(x)$$ with boundary conditions $u(0) = u(1) = 0$. We assume that the functions $g \in \mathscr C[0,1]$ and $f \in \mathscr C(\mathbb R)$ ...
3
votes
4answers
854 views

A 6 meter ladder…

A $6$ meter long ladder leans with a vertical wall and top of the ladder is 3 meters above the ground.If it slips at a rate of $2$ m/s then how fast the level is decreasing from the wall? My ...
3
votes
1answer
169 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
3
votes
1answer
2k views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
3
votes
3answers
333 views

General solution of second-order linear ODE

I am trying to look a bit deeper into the mathematics the equation of motion used in physics and engineering. I have some specific questions at the end, but please correct me if I make a mistake in ...
3
votes
1answer
369 views

Exercise from Stein with partial differential operator

I have again something from Stein-Shakarchi I would really appreciate some help with. Any references are also welcome! Suppose $L$ is a linear partial differential operator with constant ...
2
votes
1answer
27 views

Phase plots of solutions for repeated eigenvalues

I have a question with respect to phase plots of repeated eigenvalue cases. For instance suppose that one is given a matrix with the following: $$\overrightarrow{y'} = \begin{pmatrix} 3 & -4 \\ ...
2
votes
1answer
53 views

Existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$

When $t=t_0$, $f(x,t)=f_0(x)\in L^2(U)$. $t\in [0,t_0]$ and $U$ is a open subset of $R^n$.$R(x,t)$ is bounded and smooth about $x$ and $t$. I don't whether suitable the conditions is ,if not, please ...
2
votes
0answers
41 views

Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves

Consider the following PDE: $$ u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$ where $f(u)=u(u-a)(1-u), 0<a<\frac{1}{2}, \varepsilon,\gamma >0, \varepsilon\ll 1,\gamma\ll ...
2
votes
1answer
111 views

Exact Differential Equations

$M(x,y)dx + N(x,y)dy=0$ is said to be a perfect differential when $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$. Let $M_y=\frac{\partial (M(x,y))}{\partial y}$ and ...
2
votes
3answers
374 views

good book to study Differential Equations throgh geometric ideas.

When studying a subject geometric intuition is important for me. The algebra books I know, do not convey such intuition. Please, recommend books with an emphasis on geometric intuition on Ordinary ...
2
votes
1answer
181 views

Classification of operators

I have a collection of questions about the limit point/circle concept and self-adjointness that are kind of connected, so I would like to ask them in a row. Apparently, an operator that is limit ...
2
votes
1answer
117 views

:How to find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$?

question : find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$ $\frac{dx}{dt}=y+ux,\frac{dy}{dt}=x+yu,\frac{du}{dt}=u^2-1$ I dont know how to start. is this quasilinear ? edit 1: tried ...
2
votes
2answers
500 views

Expressing an oscillator as a series of ODEs

Consider an oscillator satisfying the initial value problem $u''+w^2u=0$, where $u(0)=u_0$, $u'(0)=v_0$. Let $x_1 = u$, $x_2=u'$, and transform the equations given into the form $x' = Ax, x(0)$. Then ...
2
votes
1answer
332 views

dropping a particle into a vector field

I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined ...
2
votes
3answers
134 views

Getting equation from differential equations

I have: $\dfrac {dx} {dt}$=$-x+y$ $\dfrac {dy}{dt}$=$-x-y$ and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$ I know to do this I need to decouple the equations so that I ...
2
votes
1answer
154 views

Dimensions analysis in Differential equation

Differential equation of solitary wave oscillons is defined by, $$ \Delta S -S +S^3=0 $$ How can we write this equation as, \begin{equation} \langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle ...
2
votes
1answer
986 views

Show that Bessel function $J_n(x)$ satisfies Bessel's differential equation.

here is the question: For each positive integer $n$, the Bessel function $J_n(x)$ may be defined by $$J_n(x) = \frac{x^n}{1\cdot 3\cdot 5\cdots(2n-1)\pi}\int^1_{-1}(1-t^2)^{n-1/2}\cos(xt) \, dt$$ ...
2
votes
1answer
244 views

Help with Initial value problem : $y'= x^2+ xy^2, y(0) = 0$; Picard–Lindelöf Approximation.

i need solve this: $$y'=x^2+xy^2 , y(0)= y(t_0)= 0$$ a) Compute, starting from the constant function $u_0=0$ the successive approximations $u_1,u_2,u_3$ (in the sense of the theorem of ...
2
votes
1answer
223 views

Explain Dot product with Partial derivatives in Polar-coordinates

Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas? Errors? I propose ...
2
votes
1answer
843 views

Stability of autonomous linear systems of ODEs

Let $A$ be an $n\times n$ real matrix, and let's consider the linear system of ODEs $x'=Ax$. I'm trying to characterize the Lyapunov stability of the origin according to the real part of the ...
2
votes
1answer
1k views

Boundary conditions of an elastic bar

I was following some online lecture relating to an elastic bar with length $L$ that obey the differential equation $\displaystyle \frac{d^{2}u}{dx^{2}} = f(x)$, where $f(x)$ is its own weight or some ...
1
vote
2answers
90 views

The solution of ODE $k'(x) = r(k(x))$ is infinitely differentiable if $r$ is

If there is a function $r(x)$ that is infinitely differentiable, prove that $k(x)$ is also infinitely differentiable if $k'(x) = r(k(x))$ for all $x$. I am trying to somehow prove using ...
1
vote
1answer
72 views

Approximation of $\sqrt2$ using Euler's method

Consider the differential equation $$\frac{dx}{dt}=\frac1{2x}.$$ This is a separable O.D.E. so we know how to find all of its solutions: they are of the form $$x(t)=\sqrt{t+C}$$ where $C$ is ...
1
vote
1answer
121 views

$(\partial_{tt}+\partial_t-\nabla^2)f(r,t)=0$

Hi I am trying to find the kernel of the linear differential operator $D$ $$ D\equiv\partial_{tt}+b\partial_t-a\nabla^2,\quad a,b>0. $$ We have $$ \nabla^2\equiv ...
1
vote
1answer
133 views

solving third-order nonlinear ordinary differential equation

I would like to solve: $(\frac{d^{2}y}{dx^{2}})^2+\frac{d^{3}y}{dx^{3}} \frac{dy}{dx}=0$ Thanks in advance.
1
vote
0answers
83 views

How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
1
vote
3answers
437 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
1
vote
0answers
92 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
1
vote
3answers
8k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
1
vote
1answer
134 views

Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
1
vote
1answer
101 views

Find the solution for a boundary value problem

Please, how can we find the solution of this second order boundary value problem $$-(e^{-2x}u')'-\ln(x^2+2)u= 2 e^ {-2x} - x \ln(x^2+2),\,\, x\in ]0,1[, u(0)=0,u(1)=1?$$ Or more generally, What's the ...
1
vote
3answers
26k views

What exactly is steady-state solution?

In solving differential equation, one encounters with steady-state solution. My textbook says that steady-state solution is the limit of solutions of (ordinary) differential equations when $t ...
1
vote
1answer
391 views

How to show that the geodesics of a metric are the solutions to a second-order differential equation?

On $\mathbb R^n$, let $\rho: \mathbb R^n\to\mathbb R$ be a smooth function, and $g$ be the metric given by scaling the usual flat metric by $e^{2\rho}$. I want to know how to show that the geodesics ...
0
votes
1answer
46 views

Variation under constraint

I always can't compute right.$u=u(x),R=R(x)$ and $\tau$ is constant, and $M$ is compact manifold.If $u$ is the minimizer of $$ \inf\{\int_M [\tau(4|\nabla u|^2+Ru^2)-u^2\ln ...
0
votes
1answer
64 views

Differential equation of second order (non-linear)

Is there a proper way of solving this differential equation of the second order? $$ \frac{d^2y}{dx^2}=ay^2 $$ Is it even possible?
0
votes
1answer
349 views

van der pol equation

Consider the van der Pol equation below: $(x'')+a(x^2-1)(x')+(x)=0$ I need to : Find an equilibrium point and linearize this equation near it Find solutions of the linearized equation depending on ...
0
votes
2answers
130 views

Construct the Green s function for the equation

Construct the Green s function for the equation y^''+ 2y^'+2y=0 Which boundary conditions y(0)=0 , y(π/2)=0 Is this Green s function symmetric? What is the Green s function, if the ...
0
votes
2answers
962 views

differential equations in SIR epidemic model and obtain Ro

I need to know why the differential equation system that expresses epidemic's model SIR in some texts appears: $$\frac{dS}{dt} =-\beta\frac{S}{N}I$$ $$\frac{dI}{dt}= \beta \frac{S}{N}I - \gamma I$$ ...