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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75 views
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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0answers
49 views
[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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84 views
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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55 views
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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87 views
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
$$\mathbb{L(x)(s)}=\int_0^\infty ...
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111 views
How simple is it to solve this Differential Equation
How to solve this Differential Equation?
How simple is it to solve this Differential Equation?
Any guidelines? Any hint? How to approach the solution?
Have anybody seen things like it before? ...
$$
...
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23 views
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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71 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
$$(15 − 5t)\frac{df}{dt} = (5 + 3e^{t})f + ...
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133 views
Unable to find Lipschitz constant for $y'=(t-1)\sin(y)$
Given the problem:
$$y′ = (t − 1)\sin(y),\;\;\;y(1) = 1$$
find an approximation for $y(2)$.
Give an upper bound for the global error taking $n = 4$ (i.e., $h = \frac{1}{4}$)
The goal is to find an ...
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68 views
Partial Differential Equation Eigenvalue of zero question
In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction?
That is to ...
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0answers
87 views
Problem on Hill's Equation
Show that the equation
$$
\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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49 views
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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42 views
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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95 views
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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0answers
30 views
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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68 views
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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40 views
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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41 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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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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0answers
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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55 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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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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0answers
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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23 views
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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0answers
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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69 views
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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80 views
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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44 views
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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121 views
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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0answers
77 views
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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0answers
54 views
$n$-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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0answers
172 views
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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0answers
46 views
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 ...
