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

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List of IVP known to have periodic solutions

I am looking for a list or review article describing differential equations and corresponding initial conditions which result in periodic solutions.
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63 views

Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
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35 views

What does it mean to say a differential equation is an eigenvalue problem?

My text says the following $$ \frac{\mathrm d}{\mathrm dx}\left(x^2 \frac{\mathrm dy}{\mathrm dx}\right) + \lambda y = 0,\;\;\;0\le x\le 1,\; y(1)=c\ge0$$ is an "eigenvalue problem". I don't ...
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39 views

Drawing phase portrait

This is the question in my textbook. I am a bit lost for 3 hours now. Could anyone please point me to the right direction? Let the $2 \times 2$ matrix $A$ have real, distinct eigenvalues $\lambda$ ...
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49 views

Solution to the differential equation $\frac{1}{2}\dot K-K^2+K=0$?

The solution the differential equation $\frac{1}{2}\dot K-K^2+K=0$ is given in the picture below Picture My solution $\frac{\mathrm{d} K}{2(K^2-K)}=\mathrm{d} t$ and ...
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61 views

Differential vs difference equations in mathematical modeling

I'm reading a little about mathematical modeling and I've seen some population models based on differential equations. I've also seen some (not many) that can support both difference and differential ...
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38 views

Prove that a function is locally Lipschitz

I am studying the paper "F. D. Araruna, P. Braz E Silva, E. Zuazua, Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko ...
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70 views

Periodic Solution of Riccati Equation

I want to know for which condition on "$a$" and "$k$", i.e. for which function of $a(k)$, the following Riccati equation, with the initial condition $u(0)=ia$ ($i^2=-1$), have periodic solution with ...
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26 views

Inverse Laplace Transform of $\frac{2s+5}{s^2+4s+13} $ (Check My Solution)

I have solved, just need someone to check my solution is correct. My answer is - $$2e^{-2t}\cos(3t) + \dfrac{1}{3} e^{-2t}\sin(3t)$$ Thanks
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17 views

Under which conditions the solution to a linear system of ODE has a limit?

Consider a system of the form; $$\mathbf{x}'(t)=A(t)\mathbf{x}(t)+\mathbf{f}(t),$$ $$\mathbf{x}(0)=\mathbf{b},$$ where $$\mathbf{x}(t)=(x_1(t),\ldots,x_n(t)),$$ ...
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21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
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23 views

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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49 views

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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21 views

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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32 views

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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53 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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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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71 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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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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81 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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29 views

Solution regarding Power Series and ODE's

About 4 months ago I posted Series solution to $y''-xy'-y=0$. I ran through the analysis and it appeared that I solved the ODE . The solution seemed to be ...
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58 views

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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37 views

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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28 views

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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44 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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54 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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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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16 views

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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49 views

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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97 views

Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
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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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59 views

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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27 views

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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248 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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45 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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37 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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120 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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31 views

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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48 views

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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48 views

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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47 views

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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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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46 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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38 views

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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58 views

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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61 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: ...