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

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$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
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137 views

Nondimensionalize Pendulum Equation

Equations for the dynamics of a pendulum: $A=-a\sin(\theta)-bv$ $\theta(0)=\theta_0$, $\;v(0)=\omega_0$ where $\theta$ is the angle of the pendulum with respect to it's natural resting point, $v$ ...
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41 views

Can we find $y_0$ to obtain a periodic sequence?

I way toying around with some functions and came up with the following sequence. $$ 0,1,x+1,e^x-1,-\ln(e^{x+1}+1),... $$ If $y_n=f_n(x)$, then $y_{n+1}$ is obtained using $$ \frac{dy_{n+1}}{dx} = ...
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23 views

The solvability of a Hölder ODE

The problem is as follows, I want to know whether there is a function $u\in \text{C}^{0}\left((-1,1)\right)$ but not $\text{C}^1((-1,1))$, such that $\lim\limits_{h\rightarrow ...
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91 views

Question on O.D.E

Given three parameters $L,a$ and $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ t\geq0$$ Assume that $a>0$ and $\alpha\geq 0$ We consider the ...
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72 views

why does a fractional differential equation have a unique solution?

Why must there be a unique solution to a linear constant-coefficient fractional differential equation of order $(n,q)$ with $\lceil\frac{n}{q}\rceil$ initial conditions? (All notation is as in Miller ...
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107 views

Convert PDE from cartesian to cylindrical

I am trying to convert the Navier-Stokes relation from cartesian to cylindrical. I have $3$ relations: $$\mu \left(\frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x} \right) = ...
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44 views

Ordinary Differential Equations - Sturm Liouville

Can anyone help me on the following issue? Show that between two consecutive zeros of any nontrivial real solution of $(p (x) u ')' + q (x) u = 0$ there is exactly one point of maximum or minimum ...
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48 views

Unique continuity property

Can someone told me what is :"the unique continuity property" in the following paragraph ? and what is the meaning of : .... and either $v\in E(k)$ or $v\in E(k+1)$ Please help me Thank you .
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plotting the lagrangian

From the differential equation $$\frac{\partial P}{\partial r} = \left[1+\frac{r}{\ln(1+r)}\right]D$$ I get the second-order equation $$\frac{1}{D}{P(r)}=\text{Ei}\left(2\ln(r+1)) - ...
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115 views

Periodic solutions and existence of singularity

I'm Trying to prove the following: Let $f$ and $g$ be two vector fields defined in all $\mathbb{R^{2}}$ such that $\langle f(x),g(x)\rangle$ $ = 0$ for every $x$ in $\mathbb{R^{2}}$. If $f$ has one ...
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130 views

Fourier Transform of one variable in a two variable function.

I have a function in two variables, that satisfies the following PDE: \begin{equation} \frac{x-x_0}{x-x_1}\Psi_{xx}+\Psi_{yy}=0 \end{equation} Initially I did use Fourier series \begin{equation*} ...
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53 views

Simplification of differential equation when definition interval becomes small?

Assuming the following differential equation on the interval $0<x<c$ with a rational function $f(x,c)$ $$\left(\frac{d^2}{dx^2}+f(x,c)\right)y(x,c)=0,$$ what kind of simplifications (if any) ...
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Math model - constrain GDP given different growth rates of industries

ideas needed to model national GDP given different sector growth rates subject to some contraints Given: GDP equations for $n$ industries depend on growth rates and time i.e. $g(r_1,t), g(r_2, t), ...
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85 views

Green's equation

Consider the boundary value problem (BVP) $$u'' =-f, u(O) = u''(1) = 0 \,\,\text{on}\,\, [0,1],$$ Assume $f(x)$ is a real-valued continuous function on $[0,1]$. Then, which of the following are ...
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68 views

Is this a valid application of “separation of variables”?

I asked this question over at Physics SE. I am not satisfied with the answer. At the heart of the question is this mathematical concern: Can I invoke "separation of variables" to go from this: ...
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228 views

Solving equation

Assume that $a$, $b$, $c$ and $d$ are known value integers , and $P$, $Q$, $R$ and $G$ are known value points on an elliptic curve with these equations: $$a = by+x, \\ c=\frac{1}{y}d +z , \\ P= ...
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30 views

fancy about some properties of Abel equation of the second kind in the canonical form

How can we use practically about the content in http://eqworld.ipmnet.ru/en/solutions/ode/ode0124.pdf#page=3 starting from "2. Use of particular solutions to construct the general solution." ? If I ...
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102 views

Final value theorem on coupled differential equations

Good day, I have two linear and coupled differential equations: $J_{11}\ddot{\theta_1}-n(J_{11}-J_{22}+J_{33})\dot{\theta_3}+n^2(J_{22}-J_{33})\theta_1=T_{c_1}+T_d \\ ...
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50 views

Vector field and a solution of an ODE

I have this: And my questions are : 1)what is :"The local theory of differential equations in a Banach space" 2)Why it's implies that each solution of (6) is equal to $\eta$ Please Thank you . ...
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73 views

Second order linear ODE with trigonometric coefficient

Is there a theory and a name for the second order linear ODE with trigonometric coefficient (other than the Floquet theory)? The equation in question, with $a$,$b$,$c$ periodic function containing ...
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147 views

Green function Sturm Liouville equation problem …

I need help with this new one, I don't even know how to start... $$Ly=-y''-y$$ $$y(0)=y(1)=0$$ I started doing this exercise and came to the conclusion that the result is given to us by $$G(x, ...
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61 views

Implicit differentiation of differential equation

Let the unknown cdf $F(x)$ be implicity defined by $h(F(x);a,b) := F(x)[1-a-b(1+a)] - 2 a b F'(x) x + (1+a)b = 0$, where $F(1) = 1$. Moreover, let $0<a<1$, $0<b<1$. My question is: is ...
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31 views

A journal for the article related to methods for solving Cauchy equations

I'm a postgraduate student in physics, but I have achieved interesting results in Cauchy equation. I found a reason why Adams method for solving differential equations gives rise to divergent ...
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102 views

Differential equation. Frobenius method

Let's consider the following differential equation: $t(1-t) x'' + (p-(p+2)tx' - px = 0$ a) Prove that t=0 is a singular regula point for that equation. Prove that if $p \notin\mathbb{Z}$ there are ...
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34 views

An order reduction problem.

Consider a differential equation of the form: $F(t,x,\frac{dx}{dt},...,\frac{d^{n}x}{dt^{n}})=0$ and exist $\alpha$ and $m$ fixed such that: $F(\lambda t, ...
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33 views

System of second order lineal differential equations

I have the following system: $x'' = \alpha^2 y - x $ $y'' = x- y $ I have no idea how can I start. Please give me some hint.
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32 views

Clarification on index reduction on non-linear implicit DAEs.

I have a system of DAEs (4 ODEs and one algebraic constraint) of index 2 given by the following (I'm excluding the first three ODE's as they don't have a direct effect on the index of the DAE): $$X4' ...
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30 views

Differences and derivatives.

I'm trying to figure out how to think about differentes, i.e $\Delta$. If I have an equation, $W=\Delta E$, is it at all possible to have this as a continuous function? I understand a derivative is ...
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optimal control -Taylor expansion - PDE problem

I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point. For the given control ...
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57 views

How to solve two-level Schrödinger equation using Floquet theorem?

Consider a sinusoidal driving two-level system: $$ i \left( \begin{array}{c} \dot C_1(t) \\ \dot C_2(t) \\ \end{array} \right)=\left( \begin{array}{cc} -\frac{\omega _0}{2} & ...
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59 views

Derivative methods for artifical neural networks with single hidden layer

I am trying to optimize the output of a given neural network with a single hidden layer. To accomplish this, I intend to find solve for all combinations of inputs where the derivative of the neural ...
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127 views

Schroedinger equation in cylindrical coordinates

How can one numerically solve the nonlinear stationary Schroedinger equation in cylindrical coordinates?
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69 views

When is it justified to approximate a difference equation with its corresponding differential equation?

Consider the difference equation $f_{x+1}-f_x=a(f_x)$ and the differential equation $g'_x=a(g_x)$. When and Why is it justified to say "$f_x - g_x = o(1) $ hence we can solve the difference equation ...
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128 views

“Two-speed” linear integro-differential equation

Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation: $$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
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Invariant relation in ODE

It is well known that if function $g(x)$ is an invariant relation under ODE $\dot x = f(x)$ then $\frac{\displaystyle d}{\displaystyle dt}g = \lambda g$. More precisely. Let ...
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46 views

A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$

Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$. $F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$. $P_1(F)$ is the set of all ...
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102 views

Approximating the modified Bessel’s function with a sum of exponentials

I am looking for an approximation for modified Bessel’s function $I_\alpha(f(t))$ (specially $I_0(f(t))$ or at least $I_0(t)$) with a sum of exponential functions. I mean I want to approximate the ...
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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, ...
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30 views

Green's function, stuck on this particular equation

How does one find the Green's function for a differential equation in two variables which looks like, $\frac{\partial}{\partial t} P -\omega P = j$ where $\omega$ is a $2\times 2 $matrix with ...
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92 views

How to find variance of a CIR process

CIR process is defined as follows: http://en.wikipedia.org/wiki/CIR_process I get an SDE form for d_Vt/dt, but can't proceed further.
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Pull Back (change of variables)

Let be $h:\mathbb{R^2}\rightarrow\mathbb{R^2}$ a change of variables (diffeomorphism). Let be $X$ a vector fields in $\mathbb{R^2}$ and $f:\mathbb{R^2}\rightarrow\mathbb{R}$ a continuous application. ...
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First eigenvalue of the given linear operator

I have the following question: Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$. Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by $Lu = -Du'' + Cu$, where $D$ is a positive ...
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Solving series solution near a regular singular point: $2xy''+y'+xy=0$

Disclaimer: I hate series. Burn it to the ground. Why do we even need to study this? Ok, so I haven an equation $2xy''+y'+xy=0$. The idea is to solve it using the series and euler equations learned ...
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zero stability of odes

Can anyone help with the following problem? Find the range of α for which the following method is zero stable. $$y'=f(x,y)\\ f(a)=y_0 \\ y_{n+1}-(1+α) y_n + α y_{n-1}=\frac{1}{2}h \left[(3-α) ...
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Solve I.V.P for differential using quadratic form

Solve the i.v.p for $y''+4y'+5y=0, y(\frac{\pi}{2})=1/2, y'(\frac{\pi}{2})=-2$ I solved using the quadratic form. and I got $\lambda = \frac{(-4 \pm 2i)}{2}$, which for $\lambda 1,2= 2+2i$. And then ...
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Differential Equation $y'=4|y|^{3/4}, y(0)=0$

Related questions here and here. $$y'=4|y|^{3/4}, y(0)=0$$ Let $f$ be a solution of the equation on $I$ Let $x\in I$ If $f(x)>0$, then $f'(x)=4f(x)^{3/4}$, so $f(x)^{-3/4}f(x)=4$, so ...
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150 views

Necessity for Osgood's Uniqueness Theorem

Let $\phi (z)$ be continuous and increasing function in the interval $[0,\infty)$, $\phi (0)=0, \phi (z) > 0 \, \forall z > 0$ with also: $$\lim\limits_{\epsilon \rightarrow ...
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42 views

Small Inhomogeneity of Differential Equation

Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$: $$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$ Let ...
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Entertaining video lectures of differential equation

this calculus lecture on coursera.com develops my interest in mathematics. I was very scared of math but after taking lectures of this course I just fall in love with math. I Now have Differential ...