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

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Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
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Derivation of the prolongation formula for finding symmetries of diff equations from Olver

I am having a problem with the derivation of the prolongation formula from PJ Olver's text :"Applications of Lie groups to differential equations" Page 105,106. Considering a differential equation ...
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30 views

Stability analysis of a three-dimensional system

Study the stability of the equilibrium point $(y,q,z)=(0,0,0)$. (Hypothesis: $\nu,\theta,\zeta$ are positive.) $$\begin{align} \dot{y}&=y(1-\nu -\theta -y-z+\theta y-(\nu +\zeta)q)\\ ...
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How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
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Analytical solutions of Thomas Fermi equation

The Thomas Fermi model of atoms and nuclei is used in many applications of atomic and nuclear physics. The ODE related to this model is: $$\frac{d^2}{dx^2}\phi(x)=x^{-\frac{1}{2}}\phi(x)^{3/2}$$ with ...
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25 views

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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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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41 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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Interior Products

Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product There is a line that says $\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$ where $\alpha$ is a ...
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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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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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72 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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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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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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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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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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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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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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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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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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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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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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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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99 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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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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252 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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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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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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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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50 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 ...