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

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How to calculate floquet exponents

I want to apply Floquet theory to analyse the stability of the periodic solutions for a system of differential equations. I understand the theoretical portion but how can I actually find the Floquet ...
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76 views

Pencil of conics and periodic orbits

Let $\dot{x}=P(x,y)$ and $\dot{y}=Q(x,y)$ be a quadratic polynomial differential equation. Prove that if the pencil of conics $P+\lambda Q$ contains an imaginary conic, a real conic reduced to a ...
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138 views

The ordinary differential equation $\frac{d^2y}{dx^2}-q(x)y = 0$ , $0≤x<∞$ , $y(0)=1 $, $y'(0)=1$ multiple choice question

I am stuck on the following question: Assuming $$\frac{d^2y}{dx^2}-q(x)y = 0,\;\; 0 \le x \lt \infty ,\;\;y(0)=1,\;\;y'(0)=1$$ wherein $q(x)$ is monotonically increasing continuous function,then ...
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Solution to matrix ODE $Ay'[x] + B\frac{y[x]}{x} + Cy[x] = 0$?

Does there exist a closed form solution to the homogeneous system of ODEs $$Ay'[x] + B\frac{y[x]}{x} + Cy[x] = 0,$$ where $A$, $B$, and $C$ are $n$ x $n$ (constant) matrices, and $y$ is an ...
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67 views

differential operator

I've read journal "On the Comparison of Several Mean Values: An Alternative approach" (Welch, 1951). I don't understand this expression: $$E\left(\exp\left[ \sum_t ( w_t - \omega_t ) ...
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187 views

Showing a differential equation has a unique solution in $C[0, 1]$

Show that $$F(f)(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du$$ is a contraction on $(C[0, 1), d_u)$. Deduce that the differential equation $$(15 − 5t)\frac{df}{dt} = (5 + 3e^{t})f + ...
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357 views

Asymptotic Methods - Boundary Layer Problems

I am currently studying a course in Asymptotic and Perturbation Methods and we have recently started discussing "Boundary Layer problems". It is not clear to me, however, exactly what form "Boundary ...
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160 views

Considering the predator prey model to find the range of values to be a spiral

I have the following problem: Consider the predator-prey model: $$\frac{du}{dt}=u(1-\alpha(u)-v), \frac{dv}{dt}=\rho(-1-\alpha(v)+u),$$ where $\rho$ and $\alpha$ are positive parameters with ...
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If $y(z) = C y_0(z) \int_w^z \frac{d\zeta}{y_0(\zeta)^2}$, what limit can we take in $C$ and $w$ to obtain $y(z) \to y_0(z)$?

This is Exercise 6.5 from Miller's Applied Asymptotic Analysis. The book shows that, given any solution $y_0(z)$ to the equation $$ y''(z)+f(z)y(z)=0, \tag{1} $$ a general solution is given by $$ ...
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variation of a final state due to changes in period (where the period is a parameter)

I have a simple ordinary differential equation $\frac{dx}{dt}=f(x,t,p,T)$ $x(0) = x_0$, $x(T) = x_T$ where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks! NOTE: I ...
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84 views

For what functions, “$=$” instead of “$\leq$”

Theorem: Suppose $f,g\in C(U,\mathbb R^n)$ and let $f$ be locally Lipschitz-continuous in the second argument, uniformly with repsect to the first. If $x(t)$ and $y(t)$ are respective solutions of the ...
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74 views

The importance of commuting differential operators

Consider the $\mathbb{C}$-algebra $A$ consisting of ordinary differential operators $$ \displaystyle\sum_{i \geq 0} p_i(x) \frac{d^i}{dx^i}, \ \ p_i(x) \in \mathbb{C}[x].$$ It's been known for a ...
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225 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
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2answers
863 views

Initial Value Problem: $\frac{dy}{dx} = y\sin x - 2\sin x,\quad y(0) = 0$ [duplicate]

Possible Duplicate: $dy/dx = y \sin x-2\sin x$, $y(0) = 0$ — Initial Value Problem $$\frac{dy}{dx} = y\sin x - 2\sin x,\quad y(0) = 0$$ So, I get $$\frac{1}{y-2} dy = \sin x ...
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Reconstructing paths on the sphere from the ratio of acceleration to velocity

Given a path $\gamma:[0,1]\to \mathbb C$, we can determine $\gamma$ from information about its derivatives. For example, $\gamma$ is determined by $\gamma(0), \gamma'(0)$, and $\gamma''(t)$. This ...
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1answer
611 views

Runge-Kutta ODE Solver

please keep in mind that the following is homework, I do not want answers, only help. If you are confused as to what I refer to in this question, or I don't make any sense, please refer to the link ...
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Seeking good parametrization for the homoclinic solution

Could somebody quickly provide me with a good parametrization for the homoclinic solution $$\frac{p^2}{2}-\frac{q^2}{2}+\frac{q^3}{3}=0$$ of the system \begin{aligned} \dot{q}&=p\\ ...
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Differential equations with different constants for different sub-domains

I remember that when I was studying differential equations, there was an example with solutions of the form $f(x) + C_1$ for $x>0$ and $f(x)+C_2$ for $x<0$ where $C_1$ and $C_2$ may be different ...
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215 views

Van der Pol method in a quasilinear equation with multiple fixed points within a cycle.

My question is about details of application of the van der Pol - Andronov method to analysis of quasilinear ordinary differential equations. Before formulating the question, let me first give ...
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58 views

Can a rate be proportional to a shape?

This question may be a little vague, but it has a point. I woke up this morning with an idea. Let's say I wanted to design a projectile that has a velocity proportional to its 'shape'. When the ...
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71 views

Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
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linear DE - Where did I go wrong?

I'm trying to find the general solution for $(x+2)y' = 3-\frac{2y}{x}$ This is what I've done so far: $y'+\frac{2y}{x(x+2)}=\frac{3}{x+2}$ $(\frac{x}{x+2}y)'=\frac{3x}{(x+2)^2}$ ...
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205 views

Differential Equations with Deviating Argument

Is there literature available on solving differential equations of the type $$f(x,y(x),y(\kappa x),y'(x))=0,$$ where $\kappa$ is a given constant? I know about the book Introduction to the Theory and ...
2
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83 views

ODE with irregular coefficient

Here's a simple ODE \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} I want the solution $h(x)$ to be (at least) continuous with its first and second order derivative exist only ...
2
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2answers
655 views

Does multiplying by $dt$ have any meaning?

Consider, for example, the equation $x'=x$, then it is usually solved by writing $\frac{dx}{dt}=x\implies\frac{dx}{x}=dt\implies\int\frac{dx}{x}=\int dt$ ... I know that there is a theorem in ODE ...
2
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1answer
275 views

Sturm-Liouville Eigenvalue Question

Consider the regular Sturm-Liouville Problem: $$-\frac{d}{dx} \Bigg( p(x)\frac{dv}{dx} \Bigg)=\lambda \rho (x)v$$ $$\alpha _1v(0)-\beta _1v'(0)=0$$ $$\alpha _2v(L)-\beta _2v'(L)=0$$ with ...
2
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2answers
201 views

Differential equation $d^n/dx^n f(x)=\pm k^2f(x)$

How to solve this differential equation: $$\frac{d^nf(x)}{dx^n}=\pm k^2f(x)$$ For $n=1,2,3$ and $\forall n\in\mathbb{N}$, and both signs, if this is possible.
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271 views

System of ODEs with a degenerate (?) critical point

I am not sure the name for this is really "degenerate", but consider the following system of non-linear ODEs: \begin{align*} \frac{dx}{dt} & = a(1-x)z-ex, \\ \frac{dy}{dt} & = -axy, \\ ...
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2answers
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First order ordinary differential equations involving powers of the slope

Are there any general approaches to differential equations like $$x-x\ y(x)+y'(x)\ (y'(x)+x\ y(x))=0,$$ or that equation specifically? The problem seems to be the term $y'(x)^2$. ...
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How to check if ode system is gradient system?

How do I check for a given (nonlinear) system of ODEs if this is a gradient system? Meaning how do I check the existence of a pseudo Riemann metric and a potential function? May be somebody could post ...
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136 views

Solve $x^2u''+xu'-(x^2+\frac{1}{4})u=0$ using power series

I stumbled upon this question in an old exam (I'm preparing for an exam of a course about ODEs). I didn't have much difficulty solving the Legendre and Hermite equations using power series, but this ...
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Lower bound for the eigenvalue

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
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170 views

Hermite functions and integral

Let $$ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function. Consider $$ ...
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Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$?

Consider the differential operator $D:$ $$ Du:=\frac{-d^2}{dx^2}u $$ on the function space $$ C=\{u\in C^2([0,1]):u(0)=u(1)=0\}. $$ It's not hard to find the eigenvalues and ...
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229 views

Why is del operator coordinate free?

In solving Laplace equation $\Delta u=0$, every textbook will tell you to transform $\Delta=\partial_{xx}+\partial_{yy}$ into polar coordinate form $\Delta_p$(What it looks like doesn't matter here). ...
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43 views

How to prove the only solution to f(x)=f'(x) is ce^x? [duplicate]

Possible Duplicate: Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$ I've learned from calculus that $Ce^x$ is a solution to the equation $f(x)=f'(x)$, where $C$ is a ...
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2answers
91 views

Limit of a sequence of periodic solutions

Could anyone comment on the following ODE problem? Thank you! Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be $C^{1}$ and let $X^{(n)}(t)$ be a sequence of periodic solutions of ...
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1answer
266 views

Solving Second Order Differential Equations.

How would you go about solving the following system of ODEs: \begin{align*} & x''(t) - \frac{2}{y}x'(t) \ y'(t) = 0 \ & y''(t) + \frac{1}{y} \big(x'(t) - y'(t)\big) = 0 \end{align*} Any help ...
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Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
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112 views

Showing uniqueness of solution to IVP under certain conditions

Consider the IVP $\mathbf{\dot x} = f(\mathbf{x},t)$ where $\mathbf{x}(0) = 0$, $f$ is continuous in some neighborhood of $(x,t) = (0,0)$ and $|f(x,t)-f(y,t)| \leq \frac{|x-y|}{t^\alpha}$. I would ...
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645 views

Conditions for bounded/periodical solutions of second order non-homogeneous ODE

Find all the possible values of $a$ and $b$, so that the equation: $$ \ddot{x} + a\dot{x} + bx = \sin t $$ Has only bounded solutions on $\mathbb{R}$ Has only periodical solutions In general, we ...
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1answer
176 views

Continuous variation from solution of one problem to solution of another problem

This is a differential equations question. The only thing I know about differential equations is that there can be many subtleties. Suppose that I have a function $g:\mathbb{R}^n \times \mathbb{R} ...
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150 views

Positive rotational symmetric solution for p-Laplacian

I have the the following problem and I just can't get my head around how to solve it. Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
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Where did G. W. Hill develop his relative motion equations?

The equations for relative orbital motion are commonly known as "Hill's equations" (also Clohessy-Wiltshire equations), and the citation given to G. W. Hill's 1878 "Researches in Lunar Theory" in the ...
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312 views

Implicit function theorem and consistency of a semi-explicit DAE

This may be a trivial question, but here goes: Suppose a semi-explicit differential-algebraic equation (DAE) system is defined as follows: $$ \begin{align} &\dot x = f(x,z,\theta),\qquad x(0) = ...
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270 views

Show that this is the unique solution of that Stochastic Differential Equation

Reading through a paper, I stumbled across the stochastic differential equation $ dS_t = \sigma S_{t-} dX_t $. The claim there was that $ S_t = S_0 \exp(\alpha N_t - \beta t) $ should be its unique ...
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149 views

One question on 1st-order PDE

Given a smooth vector field $\mathbf{b}$ on $\mathbb{R}^n$, let $\mathbf{x}(s)=\mathbf{x}(s,x,t)$ solve the ODE $$\dot{\mathbf{x}}=\mathbf{b}(\mathbf{x}) (s\in\mathbb{R}), x(t)=x.$$ (a) ...
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117 views

A Partial Differential Equation

What is $\frac{\partial F}{\partial y}$ evaluated at $\varepsilon=0$ for $F=\|\vec{\dot{y^\varepsilon}}\|^2$ where $$\vec{y^\varepsilon}=\frac{\vec{y(x)}-\varepsilon ...
2
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1answer
76 views

motion of a straight slender beam with a constant cross-section

I'm working on a problem and am having trouble, here's my work so far: The motion of a straight, slender beam with a constant cross-section is governed by the partial differential equation: ...
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0answers
114 views

Asymptotic stability of semi-trivial solution and existence of a nontrivial solution

Thank's amWhy! I pray to some kind soul to help me on the theory of bifurcation: In the article of Tao Peng titled: "Bifurcation Behavior of a Cohen-Grossberg Neural Network of two Neurons with ...