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

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Differential equation, eigenvalues and eigenfunctions

How does one find all the permissible values of $b$ for $-{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y$ with boundary conditions $y(0)=y(1)=0$? I assume we have a discrete set of $\{b_n\}$ where they ...
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77 views

Special forms of ODEs

In my previous question, @Gerben suggested that it is more likely that WA recognizes an ODE in"Sturm-Liouville" form. Is there a reason for this particular form being preferred to the usual ...
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117 views

Inequality of ODE solutions

Says I have two (scalar) ODE: $u' = f(u,t)$ and $v' = g(v,t)$ where Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and ...
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81 views

Differential equation with some constraints

I'd like $\alpha,\beta,\gamma$ as functions of $t$, satisfying the following conditions: $$ \begin{align} \alpha+\beta+\gamma & = 0 \\ \sin^2\alpha + \sin^2\beta + \sin^2\gamma & = c^2 \\ ...
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216 views

Half-life versus relaxation time

Question: What is the exact relationship between half-life and relaxation time? I just wanted to nail down the difference/similarity between these two concepts. I did a web search, and even found a ...
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223 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
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374 views

Differential Equations of Infinite Order

As a physicist I was playing with some QM problem and stumbled upon an ordinary differential equation of infinite order (coefficients are polynomials) that could be cast in the form: ...
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497 views

Finding the modified Green function for the Helmholtz equation

I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question: I need to find the solution to the ...
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28 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
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How to decompose connections on the complexified orthonormal frame bundle?

Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)}(E)$ the orthonormal frame bundle. I say that $P^{c}:=F_{SO(n)}(E)\times_{SO(n)} ...
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Existence and uniqueness of initial value problem in differential equation

consider the following equation: $$ y'=y^{\frac{1}{3}}, \,y(0)=0 $$ My question is how can I prove the existence and uniqueness of solutions of this initial value problem without solving the ...
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Counterexample to Peano's theorem in infinite dimension

Would you like a counter example that Peano's theorem does not apply to spaces with infinite dimension. Peano theorem: Let E be a space with finite dimension, consider a point $(t_0,x_0) \in \Re ...
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Uniqueness of the solution to a certain IVP

Let $f:[0,1]\to[0,1]$ be a strictly decreasing, continuous function with $f(0)=1$ and $f(1)=0$, and consider the following IVP: $$\frac{dy}{dt}=f(x(t))-y(t), \ \ \ y(0)=0$$ ...
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Existence of a solution of a nonlinear ODE

I have to show, that the nonlinear ODE $$u'(t)-2u''(t) u(t)=-1,\quad u(0)=1,\,u'(0)=0$$ has a unique solution $v(t)\in C^2(0,T)$ on any Interval $[0,T]$, $T>0$ and that $$\max_{0\leq t\leq ...
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44 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
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+50

second order linear ode in the complex domain

Consider $w''(z)+p(z)w'(z)+q(z)=0$ where $p(z), q(z)$ are analytic for $R\le|z|<\infty$ for some fixed $R$. Now I want to prove using analytic continuation of the solutions that the ode has one ...
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Solve by separating variables

$$\frac{dy}{dt}=e^y +1$$ I've tried: $$dy/dt - e^y = 1 $$ $$\Leftrightarrow y' - e^y dt = 1 dt$$ But I'm not sure what to do next or if I'm even doing this right!
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59 views

Homogeneous second-order differential equation with constant Wronskian

Problem Prove that if the Wronskian of an two solutions of differential equation $y''+p(x)y'+q(x)y=0$ is constant, then $p(x)$ is zero. My attempt. : Let $y_1$ and $y_2$ are solutions of given ...
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62 views

How to maximize speed of rest position approach of nonlinearly damped spring oscillator?

Inspired by comments to answer for this question: Suppose we have a system which is described by the equation $$\ddot x=-x+g(\dot x),$$ with initial conditions $x(0)=1$, $\dot x(0)=0$. If ...
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74 views

Proving the existence of a periodic solution

For a particular homework problem, I need to show that the differential equation: \begin{equation*} y^{\prime\prime}(x) - \frac{1-(y^{\prime}(x))^{2}}{1+(y^{\prime}(x))^{2}}y^{\prime} + y(x) = 0 ...
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Second order nonlinear ordinary differential equation. Help please

Can someone help me with this differential equation $$ay''(t)y(t)+2y'(t)=\left(b+\frac{c}{t^2}\right)y(t)^2$$
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non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
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Matrix differential equation MX' = AX+XB+C(t)

Here is matrix differential equation: $$ \mu \frac {dX}{dt}=AX+XB+C(t) $$ $$ X(0) = X_0 $$ Here $\mu$ is real diagonal matrix, $X$ is $m$ by $n$ matrix. $A$, $B$ are real square matrices of constant ...
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How to prove Lyapounov stability of a circle orbit?

I am trying to go through exercises in V.Arnold's book on Mathematical methods of classical mechanics. There is a following exercise there: one considers a movement in a system with central potential ...
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Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
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42 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
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50 views

first order differential equation, proof

Prove that any solution $x(t)$ of the following differential equation: $$ \dot{x}+a(t)x=f(t), $$ where $ a(t)\geqslant c > 0 $ and $f(t) \rightarrow 0$ as $t \rightarrow \infty$, tends to 0 as ...
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37 views

differential equation looks like Bessel but isn't

I have this question What I did is: $U=X(x)*T(t)$ after putting it back into the function I got $-x^2*T''/T= x^2X''-2xX'+2X $ after deviding by $x^2$ remembering to check $x=0$ I get $-T''/T= ...
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Maximizing an integral through maximum principle

Suppose that we wish to achieve $$\max\int_0^1 (1-x^2-\dot{x}^2)dt, x(0)=0, x(1)\geq 1$$ Two possible ways one can do this is by Euler-Lagrange eqn or maximum principle. Applying the Euler-Lagrange ...
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Exponential representation of picard iteration.

This is a homework question for a first course in real analysis (tiny Rudin) so I'd appreciate hints whilst straight out answers are discouraged due to academic honesty. I'm given recursively ...
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Does the solution to this ODE have a closed form?

Consider the following two initial value problems: Problem 1: $\frac{dy}{dx}=\sqrt{\frac{1}{2\cos x}-\frac{y^2}{4}}, \ \ y(0)=-\sqrt{2}$ Problem 2: $\frac{dy}{dx}=\sqrt{\frac{1}{2\cos ...
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impossible ODE using delta functions?

I'm working on the problems in the book "Asymptotic Methods of Differential Equations", by Roscoe White. It's a pretty legit book, and all the problems are quite non-trivial and very rich. However, ...
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Relationship between a class of non-linear differential equations and algebraic geometry.

I was just thinking about non-linear differential equations of a single variable, $F(f(x))=0$ that are polynomial in the derivatives of $f$. For example: $$ 2\left(\frac{d^3f}{dx^3}\right)^5 - ...
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Equilibria in Nonlinear Systems

For $x' = \sin x$ and $y' = \cos y$, (a) Find all of the equilibrium points and describe the behavior of the associated linearized system. The equilibrium points I found are $(m\pi, ...
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ODE and domains of existence and uniqueness

Find the one parameter family of integral curves and state the domains of definition , existence and uniqueness ( validity ) of the solution. Use the existence and uniqueness theorem to substantiate ...
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182 views

Find all of the equilibrium points and describe the behavior of the $x' = sin(x), y' = cos(y) $.

Find all of the equilibrium points and describe the behavior of the $$x' = \sin(x), \quad y' = \cos(y) .$$ It has been a while since I took DE...Do we first need to set $x' = y'$ to solve for their ...
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39 views

Periodic solutions to ODEs

I have the second order ODE $\dfrac{d^2x}{dt^2}-\bigg(\dfrac{dx}{dt}\bigg)^2 + x^2 - x = 0$. I have transformed it into a plane autonomous system, and then the question asks: By considering ...
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34 views

Solving $(y'')(y'-y)(y''+4y)^2=11e^x-\sin(2x)$

I am trying to solve this equation. I have solved the homogeneous part as: $y(x)=ae^x$ or $y(x)=ax+b$ or $y(x)=a\cos(2x)+b\sin(2x)$ correct me if I am wrong, but stuck with the particular part. Can ...
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92 views

Maximum principle question in partial differential equation

Problem is Let $U$ be a bounded domain in $\mathbb{R}^{n}$ and $\vec{b} : \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \to \mathbb{R}$ be continuous. Show that there can be at most ...
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Constant of motion in a high dimensional analogue of the Lotka-Volterra system.

Suppose I would extend the Lotka Volterra system to the the $n$-dim first order ODE \begin{eqnarray*} \dot{x}_{1} &=& x_1(x_2-\alpha_1) \\ \dot{x}_{2} &=& x_2(x_3-\alpha_2) \\ ...
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How do I solve this question without solving for the functions?

The problem goes as follows: $$\begin{aligned} \frac{d y_1}{dt} &= -ay_1 \\ \frac{d y_2}{dt} &= -by_2 -\frac{dy_1}{dt} \\ y_1(0)&=M \\y_2(0)&=0 \end{aligned}$$ where ...
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Spectrum of the Orr Sommerfeld equation

The Orr Sommerfeld equation is as follows $$\psi''-k^2 \psi - \frac{U''}{U-c}\psi=0$$ where $\psi(y)$ is a complex valued function on $[0,2\pi]$ satisfying Dirichlet boundary conditions ...
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Eigenfunctions.

I have the following ODE: $$y''-2xy'+2\alpha y=0$$ whose solution $y(x)$ may be recursively represented as: $$a_{n+2} = \frac{a_n(2n-2\alpha)}{(n+2)(n+1)}$$ I have found the eigenvalues to be ...
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35 views

value of $\alpha$ for which the infimum of the set is greater than or equal to $1$

Let $y(x)$ be the solution of the differential equation $$\frac{d^2y}{dx^2}-y=0$$ such that $y(0)=2$ and $y'(0)=2\alpha$. Find all the values of $\alpha \in [0,1)$ such that the infimum of the set ...
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34 views

Hopf Bifurcation of Reaction-Diffusion System

I'm considering the following reaction-diffusion system: $ \frac{\partial u}{\partial t} = f(u,v)+ D_1 \frac{d^2 u}{dx^2} $ $ \frac{\partial v}{\partial t} = g(u,v)+ D_2 \frac{d^2 v}{dx^2} $ where ...
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Singular Solutions of this Equation?

How would I find the singular solutions of this equation: y = $ce^{x^2}$ + $ce^{\sin x}$ (where $c$ is a constant). It should be $x^2$ if anyone gets confused by the first part of the equation. ...
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Translation invariance and finite dimension imply smoothness

Let $X$ be linear subspace of $C(\mathbb R)$, the set of continuous functions on $\mathbb R$, which is closed under translations, i.e., if $f\in X$ and $h\in\mathbb R$, then $\tau_h f\in X$, where ...
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58 views

On a specific non-linear partial differential equation

Given an $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$ and the functions $h_i: \mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\dots,l$, we would like to find a solution of the following equation: ...
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27 views

ODE: Integration Factor

Given $(4xy)dx$+$(x^2-3y)dy=0$ I conclude that an integration factor is needed. At the end of the process I get to $\ln(u(y))=-0.5\ln|y|$. Am I allowed to use $u=1/y^{0.5}$ as an integration factor, ...
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How to properly prepare for a gradute level PDE course using Evans and Strauss ' book

For my undergrad background , I have cal 1-3, linear algebra , 1 semester of ODE, 1 semester of real analysis never have any PDE before. Thus I know this background is hardly enough to do well in a ...