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

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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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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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48 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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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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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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247 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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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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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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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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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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61 views

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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General solution to a differential equation with an initial condition

How can I find the general solution of this DE? where $D$, $R$, and $I$ are constant parameters. $$\frac{D}{2}\frac{\ d ^2 y}{\ d x^2 \,} + {(x-RI-D)}\frac{\ d y}{\ d x} + (1+\lambda){y}{} = 0$$ ...
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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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312 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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67 views

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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540 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 ...