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

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2
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167 views

Method of undetermined coefficients for the input functions associated with the unit step

I am trying to solve a second order non-homogeneous differential equation where $x(t)$ has $u(t)$, the unit step as a part. i.e. $ x(t)= f(t)u(t) $ I know how to 'guess' the particular solution for $ ...
2
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0answers
158 views

Unusual jump condition for Green function

This question is related to a previous question I posted a while ago. Imagine that I'm computing the Green function of a linear operator $L$, such that: $$LG(x,s)=\delta(x-s).~~~~~~~~~~~(1)$$ Now, ...
2
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594 views

How to apply Duhamel's Integral

I found one good procedure for solving the simple system of two equations with reducing on Duhamel's Integral, but I have problem to apply the same procedure on system with four equations. Let's see ...
2
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0answers
45 views

I have an infinite solution to an ODE even though it has only a regular singular point

I have the ODE: $\displaystyle y''(x)+\frac{y'(x)}{x+1}+y(x)=0$ I know that this has a regular singular point at $x=-1$, as $(1+x)^{-1}$ has only a first order pole, and $1$ has no pole at all, and ...
2
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79 views

Stability of limit cycle

What can be said about the stability of the limit cycle for $r=1$ of the equation $$\dot{r}=(r^2-1)\cdot (2 r \cos(\phi) - 1), \dot{\phi}=1?$$ This is a problem posed in Arnol'd's book on ODEs. ...
2
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70 views

Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff ...
2
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49 views

Solving more complex diferential equations

I've come up with this implicit equation $ (y')^2(2x-2x^2+2y^2)+(y')^2=1 $ and I'd like to find the function $y(x)$ (so that it's definition doesn't contain it's derivative). Only thing I've been ...
2
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148 views

Show that this orbit has a zero Lyapunov exponent

I'm using J.Meiss -Differential dynamical systems, and have some trouble to understand a proof about Lyapunov exponents. We have a dynamical system $$ \dot{x} = f(x), $$ with the corresponding flow $ ...
2
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225 views

Hodograph transformation and implicit solution of a non-linear PDE

I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
2
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749 views

Existence of a unique solution with given initial value problems.

Directions: Find an interval centered about $x = 0$ for which the given initial-value problem has a unique solution. $$(x - 2)y'' + 3y = x$$ Initial values: $y(0) = 0,\,\,y'(0) = 1 $. My answer ...
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87 views

van der pol and liapunov

i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated. a) show that the second-order differential equation for ...
2
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101 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
2
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269 views

Floquet's Theory, Hills Equation

Let us examine Hill's equation $\ddot x+Q(t)x=0$, where $Q$ is piecewise continuous and with a period $T$. Let $\mu_{1,2}$ be the multiplicators. Let $\lambda$ be the characteristic exponent. How can ...
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35 views

Property of first order differential equation

I need help with following exercise: Let $f$ be real function in $R$ of class $C^1$ and $f(r)=r$. Show that if $f'(r) \lt 1$ then no solution of the equation $x'=f(x/t)$ is tangent at $0$ to ...
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35 views

Pure differential equation whose solution is a siluroid?

I am trying to find a differential equation for the siluroid that DOES NOT contain explicitly $\theta$, $\sin\theta$, or $\cos\theta$, but only $\rho$, $\dot\rho$, $\ddot\rho$. The siluroid equation ...
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62 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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86 views

Unusual 2nd order inhomogeneous equation..

For some research Im doing, I've derived an equation of the form below for $C(r)$ $$C'' + \frac{2}{r}C' = W + \frac{f}{C}$$ Or, if you prefer, $$CC'' + \frac{2}{r}CC' - W\cdot C = f$$ This has the ...
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189 views

Construction of a system differential equation from the projected 2D solution curves

! A 3 variable system.I've been give the behaviors on the 2 projected planes. How do I arrive at the final 3d curve? My question is a very simple one.I have a system of 3 variables:- X,Y,Z.I have ...
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50 views

Approximate Differential Equation?

Let $x \in \mathbb{R}$ be a variable and $c\in\mathbb{R}$ a parameter. Also, let $f(x,c)$ be a function dependent on $x$ and $c$. Furthermore, define a Differential Equation which is solved by the ...
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46 views

Generating Function for rational sequence

I'm trying to compute the generating function for the function defined for $1\le N < \beta$, $C_N = \frac{\beta}{N(\beta-N)}$. I think my math so far is correct, but I don't know how to solve the ...
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73 views

closed form solution of an ODE

I have a problem with finding the closed form solution of the following ODEs. Closed form here means that the solution can be presented as integrals/ power series. Here is the ODE : I only consider ...
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103 views

Invariant Subspaces and Differential Equations

Given I'm given a marginally stable system, $\dot{x}(t)=Ax(t)$, where$A=\begin{bmatrix} -1 & -10 & -10\cr 1 & 0 & 0\cr 0 & 1 & 0 \end{bmatrix}$, and $x(0)=x_o$.The eigenvalues ...
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116 views

ODE: continuous dependence on parameters

Is it true that the solutions of the problem: $$\begin{cases} \frac{\text{d}}{\text{d} s} [s^{2-2/N} u^\prime (s)] + \frac{\lambda}{c_N^2}\ u(s)=0 \\ u(\bar{s})=1\\ u^\prime ...
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111 views

phase portrait question

My DE course uses an online homework service to distribute and collect homework. One of the problems in this set is to furnish an autonomous DE consistent with the phase portrait below: I came up ...
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45 views

a nonlinear ode with an quotient

I'm curious about the nonlinear ODEs with an quotient including dependent variable : $$y''(x)+\frac{Ay(x)x}{1+y(x)}+Bx=0$$ Could you give me a clue on solving this equation explicitly?
2
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62 views

What's the shooting algorithm for the mass-spring problem (ode)?

I have the following problem : $$ \begin{aligned} \frac{d x(t)}{dt} &= y(t)\\ \frac{d y(t)}{dt} &= -x(t)+y(t)(1-x(t)^2)+u(t) \end{aligned} $$ with the initial condition $(x,y) = (0,0)$. Those ...
2
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239 views

Condition so that $y''+p(x)y'+q(x)y=0$ can be converted in a ODE with constant coefficients

I have to find a necessary and a sufficient condition for the functions $p$ and $q$ so that the linear differential equation : $y''+p(x)y'+q(x)y=0$ can be converted in a linear differential equation ...
2
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224 views

A dynamic Stackelberg game - general characterization

my question is about general representation of a dynamic Stackelberg game which is played in continuous time. We have maximization problems of two agents who play this game. Agents are 'Leader' and ...
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58 views

A differential equation with “$xy$” term

How to find the general solution of $$c_1xy+c_2xy\frac{dy}{dx}+c_3=0$$ Where $c_1,\ c_2,\ c_3$ are constants
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54 views

2nd order ordinary differential equation problem

I would like to solve this differential equation and I am looking for the technique/answer Consider a function $f=f(r)$ The differential equation is $$ f'' + ...
2
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0answers
79 views

A family of functions

Does there exist an infinite family of functions which satisfy $|f^\prime(x)|=1$ and $f(1)=f(-1)=0$? where a) $f\colon \mathbb{R}\to \mathbb{R}$ b) $f$ is a complex function defined on some open ...
2
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0answers
294 views

How to convert a hologram into an image?

Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...
2
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0answers
114 views

Positive eigenvalues in differential-algebraic equations not appearing in time-domain simulation

I am solving a system of equations derived from power system applications. It consists of index-1 differential and algebraic equations in the form: $$\dot{x}=f(x,y) \\ 0=g(x,y)$$ To get the ...
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886 views

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 ...
2
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72 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 ...
2
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135 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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68 views

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 ...
2
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0answers
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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0answers
180 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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0answers
288 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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151 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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44 views

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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0answers
29 views

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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0answers
83 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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69 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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176 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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0answers
59 views

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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0answers
35 views

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\\ ...
2
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71 views

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 ...
2
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0answers
177 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 ...