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

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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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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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56 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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103 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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60 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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31 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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262 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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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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127 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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38 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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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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52 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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50 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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51 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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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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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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Parabolic PDEs Maximum Principle

Consider diffusion equation for t>0 and $ \boldsymbol{x} $ in a bounded doman $ D$ in $\mathbb{R}^n$, and a given scalar field $a(\boldsymbol{x}) >0$ that is uniformly bounded and continuously ...
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37 views

Power Series for Original Differential Equation

The question: $y"+x^2y'+2xy=0$ I continue to get the incorrect answer and not sure why. I changed my indices around to make x^n all throughout and that's where the trouble starts. My answer ...
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perturbation question

I'm a little stuck with a problem and I was hoping that you guys could help. Question: A projectile is fired up from the earth with an initial velocity of $v_0$ upwards. Accounting for air resistance, ...
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Differential equation question on oscillations

For any constant $k$, show that any solution of the equation$$x^2y''-xy'+(e^x-kx^2\sin x)y=0$$has infinitely many positive zeros. From what I've learnt, I managed to convert the equation into its ...
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Stochastic input to LTI systems

Papoulis stochastics processes book question 9-6 says: Show that if $R_\nu(t_1,t_2)=q(t_1)\delta(t1-t2)$ and $\mathbf{w''}=\mathbf{v}(t)U(t)$ and $\mathbf{w}(0)=\mathbf{w'}(0)=0$ then ...
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(Sobolev space) A star domain in $\Bbb R^n$

We know the theorem: If $\Omega$ is a star domain in $\Bbb R^n$, then $C^{\infty}(\overline{\Omega})$ is dense in $W^{k,p}(\Omega)$. It means that ...
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371 views

Torricelli's Law

A water tank is made by rotating the graph of the function x=g(y) about the y-axis. The volume of the tank at height y is then given by $V(y) =\pi\int_{0}^{y}g(u)^2 \, du$ A hole is made in the ...
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Hypergeometric Function Differential Equation

Is there some nice obvious way to see that the hypergeometric function $$_2F_1(a,b;c:z) = \sum_{i=0}^\infty \tfrac{(a)_n(b)_n}{(c)_n}\tfrac{z^n}{n!}$$ should satisfy the differential equation ...
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Evaluating an integral $\int \sqrt{x^{2k}+x^{-2k} - 1} dx$ (encountered in pursuit problem)

Evaluate the indefinite integral $$\int \sqrt{x^{2k}+x^{-2k}-1} dx$$ where $k\in \mathbb{R}$. Source of inspiration: The pursuit problem of fox on rabbit. Rabbit with speed $v_R$ starts from origin ...
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632 views

Checking my work on a Lennard-Jones potential problem in differential equations

The Lennard-Jones potential is $$U(r) = \left[ \left( \frac{\rho}{r} \right)^{12} - \left( \frac{\rho}{r} \right)^6 \right]$$. What is the equilibrium distance? OK, so I know that the equilibrium ...
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174 views

What does $d/dx$ actually mean?

I'm starting to learn about differential equations, and I'm having trouble mentally adjusting to working with differentials as separate quantities. (I took calculus in high school and college but I ...
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120 views

Solving $ \mathbf{y'}(t) = \omega(t) + \frac{1}2\omega(t) \times \mathbf{y}(t) $

How to solve: $$ \mathbf{y'}(t) = \omega(t) + \frac{1}2\omega(t) \times \mathbf{y}(t) $$ or equally: $$y_1′(t) = \omega_1(t) + \frac{1}2(\omega_2(t)y_3(t) - \omega_3(t)y_2(t))$$ $$y_2′(t) = ...
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167 views

Weak/Variational Gronwall type inequality

I came across the following weak differential inequality while looking through F.Otto's paper on $L^{1}$ contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - ...
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63 views

Differential Equation Logistic Curve

NOT A DUPLICATE - see comments below I have to find P1 where the other question does not. Also the A = some function equation is different from mine. I get this far and realize if I substitute ...
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Supersingular elliptic curves- Invariant differential exact proof question

I'm writing a minor thesis about different criteria of supersingularity and I wanted to show the following from Husemöller's Elliptic Curves [Prop. 13.3.8]: An elliptic curve $E$ in characteristic ...
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149 views

What Happens At An Equilibrium Point For An Autonomous First-Order Differential Equation.

Let $\frac{dx}{dt} =f(x)$ be an autonomous first-order differential equation with equilibrium point at $x_0$. a) Suppose $f'(x_0) = 0$. What can you say about the behaviour of the solution near ...
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Why do we take 2 derivatives of the right side of a heterogeneous ODE when using the method of undetermined coefficients?

Let g(x) be the right side of a heterogeneous ODE. Why do we take 2 derivatives of g(x) when using the method of undetermined coefficients? g(x), g'(x), and g''(x) is used to guess the form of the ...
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Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } ...
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When solving PDEs is there an alternative to interpolation for out-of-grid point?

I'm numerically solving a PDE where the space domain is huge. So, I often need to interpolate to get out-of-grid points needed by the finite difference algorithm. As a result, I've a lot of numerical ...
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A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
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Solving an eigenvalue problem on the open unit rectangle

Let $\Omega=(0,1)\times(0,1)$ and consider the boundary value problem $$\begin{cases}\Delta^2u=f\\ u(x,y)=\Delta u(x,y)=0,& x,y\in\partial\Omega \end{cases}$$ I want to solve this boundary value ...
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Ordinary Differential Equation and graphs theory?

Is there any application of Ordinary Differential Equation in graphs theory?
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36 views

eigen problem for direct scattering method

Consider the KdV equation $$u_{t}+6uu_{x}+u_{xxx}=0$$ with initial condition $$u(x,0)= \begin{cases} 1 &\text{if } x \in [-1,0] ,\\ 0 &\text {if } x \in ...
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Establishing bounds of differential equation using a maximum principle

I would like to establish that the solution of $$-\epsilon u''_\epsilon+b(x)u'_\epsilon=f(x)$$ satisfies $$||u^{(k)}_\epsilon||\leq C(1+\epsilon^{-k/2}),$$ where $b,f\in C^4(\bar\Omega)$, $b(x)\geq ...
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88 views

Exact Differential Equations of Order n?

A second order ode $Py'' + Qy' + Ry = 0$ is exact if $$(Ay' + By)' = Ay'' + (A' + B)y' + B'y = Py'' + Qy' + Ry = 0$$ How can one cast the analysis of this question in terms of exact differential ...
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110 views

solve $axy''-by'+cxy=0$ step by step

Solve $$axy''-by'+cxy=0$$ step by step I know the solution is $$y=k_1x^{u}J_{u}\left(\sqrt{\frac{c}{ a}}x\right)+k_2x^{u}Y_{u}\left(\sqrt{\frac{c}{ a}}x\right)$$ Where $k_1,k_2$ are arbitrary ...