# Tagged Questions

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

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87 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

Assuming $$\frac{d^2y}{dx^2}-q(x)y = 0,\;\; 0≤x<\infty ,\;\;y(0)=1,\;\;y'(0)=1$$ wherein $q(x)$ is monotonically increasing continuous function,then which one of the following is true. (a) ...
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### Solve equation by Fourier Series

Given the equation $\Omega = a(x) \, + \langle \omega \mid \nabla_x \log \lambda(x) \rangle,$ where $x \in \mathbb{T}^n, \, a(x) > 0, \, \Omega > 0, \, \omega \in \mathbb{R}^n.$ I have to ...
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### [High Scool/College DE]Synchronization of metronomes

I'm not sure about the level of Discipline, since i'm from a country that does education differently than US. Basicly i'm working on an assignment that requires me to learn something that isn't in our ...
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### Lipschitz Questions

I want to ask one general question, and after that I would like to know if my method is correct (for determining whether a function is Lipschitz with respect to y) Is the following statement true? ...
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### Behaviour of solutions of a differential system at bifurcation values

What are the local and global behavior of solutions of $$\left\{\begin{array}{ll}r'&=r−r^2\\ \theta'&=(\sin\theta /2)^2+a\end{array}\right.$$ at all bifurcation values?
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58 views

### Find the limit of the following integral

I need help finding, $$\lim_{t\to\infty}\int_0^t \exp((t-s)A)g(s)\,\mathrm{d}s$$ when $$\lim_{t\to\infty} |g(t)|=g_0$$ Here A is a nxn matrix, whose eigenvalues satisfy $$\Re(\alpha_j)<0$$ and ...
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75 views

### Ordinary Differential Equation in three variables

I have the following ODE: $\frac{dy}{dz} + \frac{y(x+y)}{(y-x)(2x+2y+z)} = 0$ where z is a function of x and y, i.e. $z(x,y)$. For an example in two variables, I used integrating factor but I ...
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### Laplace's transform

Let $$x:[0,\infty) \to \mathbb{C}^n$$ such that $$|x(t)|\leq Me^{\alpha t}$$ for some constants $M \geq 0$ and $a\in \mathbb{R}$. Then the Laplace transform \mathbb{L(x)(s)}=\int_0^\infty ...
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### How simple is it to solve this Differential Equation

How to solve this Differential Equation? How simple is it to solve this Differential Equation? Any guidelines? Any hint? How to approach the solution? Have anybody seen things like it before? ...  ...
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### What's the physical meaning of the boundary value problems at resonance?

Many papers deal with the boundary value problems at resonance. But how to understand the problems at resonance? Why do they talk about these kinds problems? What is the physical meaning?
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### 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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133 views

### Unable to find Lipschitz constant for $y'=(t-1)\sin(y)$

Given the problem: $$y′ = (t − 1)\sin(y),\;\;\;y(1) = 1$$ find an approximation for $y(2)$. Give an upper bound for the global error taking $n = 4$ (i.e., $h = \frac{1}{4}$) The goal is to find an ...
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68 views

### Partial Differential Equation Eigenvalue of zero question

In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction? That is to ...
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87 views

### Problem on Hill's Equation

Show that the equation $$\frac{d^2\space y} { d\space x^2}+ y\sin^2 (100t)=0$$ has only bounded solutions. I was trying to prove $|y(1)(p) + y(2)(p)|< 2$ where $y(1)$ and $y(2)$ are $2$ ...
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### Applying the Lagrange Euler Formulation

I was doing my tutorial on Lagrange-Euler formulation for robotic systems when i came across a slight problem. Referring to the picture in the link, I would like to know if my answer (equation 1) ...
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### How can we apply differential equations to optics.

Differential equations in itself is a very complex topic. I read this article on a website that we can apply differential equations to optics and something like brachistochrone problem. What is this ...
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### A question about nonlinear ODE and chaos

I'm just being curious, but is it true or false, that every 3 dimensional nonlinear ordinary differential equations, after rightful parameterizing, can become chaotic? If not, what kind of 3-D ODE can ...
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### What is the definition of Cauchy function associated with the differential or difference equations?

What is the definition of Cauchy function associated with the differential or difference equations? Where can I find the details?
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### Poisson rate regression for grouped data: How to derive alpha and beta

A study of patients’ survival was classified by sex (female or male) with follow-up of patients until the patient died or the study ended. We have the following information: $y_1$ - Number of deaths ...
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28 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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38 views

### Differential Inequality Conditions to Determine Exponential Growth/Decay

I'm kind of new to differential equations and I was looking at differential inequalities. I was wondering if I had a second-order differential inequality of the form $f''(x) + af'(x) + b \leq 0$ where ...
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107 views

### Equilibrium point of a certain ODE system

Suppose given a system of ODEs $x' = sx^{r} - x\left(sx^{r}+ty^{r}\right) ;$ $y' = ty^{r} - y\left(sx^{r}+ty^{r}\right) ,$ such that $x+y\equiv 1$ and $s,t,r\in\mathbb{R}_{>0}$. The points ...
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### Compute Rayleigh quotient for ODE

I am trying to find Rayleigh quotient for this equation: $u''(r) + [\frac{1-4n^2}{4r^2} + \lambda - 2n\beta -\beta^2r^2]u(r) = 0$, where $0 \le r \le 1$. Is there any way to compute eigenvalue ...
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### 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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90 views

### damped harmonic oscillator driven by a stochastic momentum (not force)

Could you give references for solutions or solutions to the following problem: Given: damped harmonic oscillator driven by stochastic force of very short duration (= stochastic momentum). Find: ...
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17 views

### Is it possible to further simplify the following equation?

Is it possible to write the following equation in an even simpler form? (In other words does this have any specific implications on the form $\vec f(\vec x)$ can take?) {\partial f_j(\vec x)\over ...
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30 views

### Monge-Ampere equation

I'm considering the Monge-Ampere equation in $\mathbb{R}^n$: $\mathrm{det}(D_{ij}u)=f$. I know that its linearized coefficient matrix is $\mathrm{cof}(D_{ij}u)$, i.e. the co-factor matrix of the ...
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44 views

### Stochastic differential equations and experimental data

If we have a set of experimental data: $$X=\{x_1,x_2,\ldots,x_N\}$$ is it possible to write down an equation of the kind: $$dx(t)=b(x(t))\,dt+\sigma(x(t))\,dB(t)$$ describing the process from which ...
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57 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 ...
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51 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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48 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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### 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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115 views

### maple code for exp-func. for solving PDE's & non-linear ODE's?

How can I create the Maple code using exponential-function solving the equation below? $u_t = \gamma u_x+6u(u_x)^2+(3u^2-1)u_{xx}-u_{xxxx}$ $u_t =u_{xx}-u^3+u,$ \$\alpha u''(x) = \beta ...
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175 views

### Solving the boundary value problem by means of Galerkin method

I have a task which should be solved with Galerkin method: $$y''-0.5x^2y+2y=x^2 \\ y(1.6)+0.7y'(1.6)=2 (1)\\ y(1.9)=0.8 (2)$$ I already solved it with other methods so the correct answer I know, ...
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51 views

### Question about differential equation notation

I'm trying to read the paper "Particle flow for nonlinear filters with log-homotopy" by Daum and Huang. ( http://144.206.159.178/ft/CONF/16415230/16415269.pdf ) As ...
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### EDP with complementary terms

I am considering a problem of a two-dimensional ODE involving Karush, Kuhn and Tucker conditions on one of the unknowns. After a few algebraic manipulations, I end up having to solve the following ...
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120 views

### how does solution of simultaneous DE of first order relate to total DE

As far as I've understood $${dx \over P} = {dy \over Q} ={dz \over Q} \hspace{2 cm} (1)$$ Gives system of curves, $v= 0$ and $u=0$ be it's two solution. The solution of $(1)$ is the intersection of ...
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130 views

### Phase Plane Analysis

Classify the fixed point at the origin and sketch an accurate phase portrait for the following system: $$dx/dt = 36x-16y$$ $$dy/dx = -3x+28y$$ Am I correct in thinking that I need to write these two ...
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114 views

### About the Legendre differential equation

Consider the Legendre differential equation $$(1-x^2) y'' - 2xy' + n(n+1)y = 0$$ Then its solution is given by $$y = c_1 P_n (x) + \text{an infinite series}$$ In fact \$y = c_1 P_n (x) + c_2 Q_n ...
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### Collision of eigenvalues of a linear ODE (Krein collisions)

I am trying to understand the so called Krein collisions in Hamiltonian mechanics but I shall formulate the question in a rather general way. Suppose we have the following linear ODE: \$ \dot{v}= ...
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117 views

### Intuition for PDE Change of Variables

The algebraic manipulations for changing variables in PDE/ODE problems are often very simple once you know the transformation to use (at least at my level it's just applying the chain rule carefully). ...
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118 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 ...
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105 views

### How to derive to inverse z transform of $\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$ from Laguerre differential equation?

How can I derive the inverse z-transform of: $$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$ If Maple is not the way, how to derive manually? With Maple code I encounter some problems ...
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### What is the correct differential equation for the Laguerre function?

I would like to derive the correct Laguerre function from the differential equation but the differential equations seems different from the original one. What is the correct differential equation and ...
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### homoclinic solutions to an autonomous fourth-order ODE

Let $b, l > 0$ and $\mu > 2$. Let $F \in C^2([0,\infty))$ and $f = F'$ with $f(q)/q \to 0$ as $q \to 0^+$, $f(q)/q$ increasing, and $qf(q) \geq \mu F(q) > 0$ for $q > 0$. Consider the ...
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### solving two systems of equation implicitly

I have been trying to solve the following two systems of equations simultanously and I'm very hesitant on how to go about it. Whether I need predictor-corrector methods, if I need to linearize the ...
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### Literature on Riccati equations (algebraic and differential)

Advise me please some book on algebraic and differential Riccati equations: I'm interested in such questions as theorems of existence, uniqueness and extendibility of solutions of differential ...
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### $n$-th derivative of the prolate spheroidal function

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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### Semi implicit integration - stability issues

I am trying to decide whether to use semi-implicit integration vs. explicit integration (particularly Position Verlet over Semi implicit Euler). Although the Verlet approach is widely used and is ...
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### Implications of given solutions

This has been solved! Thanks to everyone who read and thought about it Suppose lines of the form $(x_0,y)$ and $(x,y_0)$ for any given $x_0,y_0\in \mathbb R$ are solutions to the system of ...
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