Questions tagged [ordinary-differential-equations]
For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.
44,471
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A continuous differential equation with no solution
I am studying the following counterexample.
But I don't understand where the fact that space is infinite dimensional is used. What fails in an infinite dimensional space?
- 1,070
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A first order linear differential question that may have only the trivial solution $y = 0$.
Below is a problem I made up. I expected the differential equation to have
a unique non-trivial solution. However, it did not. Is my solution wrong?
Problem:
Solve the following differential equation:...
- 4,112
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0
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17
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find numerical solution of an integral equation
Let $f(u)=\int_0^u f(x)g(u-x)dx+h(u)$ where $g$ and $h$ are known, continuous function and $f$ is unknown and let $a=u_0<\dots<u_n=b$. How can I calculate $f(u_i)$ numerically? I thought about ...
- 351
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Velocity Field Connecting 2 points
Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t).
Now there is a bounded, simply connected open subset $\Omega\subset \mathbb{...
- 138
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1
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40
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How does the classification of these fixed points changes with $\epsilon$?
Write the second-order differential equation $\ddot{x}+2\epsilon\dot{x}+sin(x)=0, \epsilon\geq 0$, as a pair of coupled first-order equations. Find all its fixed points, and determine how the ...
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$w''(x) + \gamma w'(x) = \lambda e^x w(x), \quad 0 < x < 1$ Prove that any eigenvalue $\lambda$ for this problem must be real-valued. [closed]
$w''(x) + \gamma w'(x) = \lambda e^x w(x), \quad 0 < x < 1$
with boundary conditions:
$w'(0) + \gamma w(0) = 0, \quad w(1) = 0,$
where $\gamma$ is a real-valued constant satisfying $\gamma$ > ...
- 151
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58
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solving partial differential equation of 2 variables - particular part of the solution
I'm trying to solve the differential equation:
$a \frac{\partial u}{\partial t } = b \frac{\partial^2 u}{\partial z^2} - G$
with B.C.:
$u(z=0) = 0, u(z=1) = cos(t)$
and I want to find the solution ...
- 27
-2
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Finding Transfer function, eigenvalue(s), time constant, damping ratio, damped natural frequency, and gain of differential equations
Control systems class, first homework and I am already stuck.
Asked to find the Transfer function, eigenvalue(s), time constant, damping ratio, damped natural frequency, and DC gain of various first ...
1
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1
answer
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How to find the upperbound/(verify it) of the solution to this differential equation. Thanks in advance!
Let $y=f(x)$ be the solution of the differential equation $\frac{d y}{d x}=\frac{x+2 y^2}{2+y^2}$ where $f(4)=0$ then which of the following are true. Options are
(A) $f^{\prime}(x) \geq 2 \forall x&...
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Laguerre Polynomials and complex arguments
I was trying to solve this differential equation:
$$
\frac{\partial^2 \phi}{\partial t^2} - c^2\left( 1- \frac{a}{r}\right )\nabla^2 \phi = 0
$$
$$
a \in \mathbb R, \; \; \; t > 0, \; \; \; r > ...
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Boundedness of solution to linear homogeneous second-order ODE
Consider a second-order linear homogeneous ODE, also known as the steady 1D Schrödinger equation
$$
x''(t)+g(t)x(t)=0,
$$
where $g(t)$ is a $C^1[0,+\infty)$ function and has a positive lower bound (...
- 106
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Runge-Kutta 4 code: no 4th order improvement
In short: in a specific physics problem, my Runge-Kutta 4th order code doesn't produce the desired and expected fourth order improvement when I reduce the iteration step size. What is the cause?
In ...
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Elementary change of variable in an ODE that makes me doubt
I am quite ashamed of the level of my question, but I have this ODE -- it is a simplified version -- and my concern is only with the left hand side:
$$ \dot v=kv$$
which I need to rewrite in terms of ...
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Can the differential operator be the result of dividing a derivative by the function itself?
I'm confused about some differential calculus in a textbook, and I'm not sure if I'm confused or the book has a mistake.
In Applied Hydrology by Ven Te Chow et al on page 9 we have:
$$k\frac{dQ}{dt} + ...
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45
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Find a condition that makes this differential equation homogeneous
Let $\alpha$, $\beta$ and $\gamma$ be non zero real numbers. Consider the differential equation
$$x’ = t^{\gamma - 1} H\left(\frac{x^\beta}{t^\alpha}\right)$$
where $H : \mathbb{R} \rightarrow \mathbb{...
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ODE y'=f(x)y: When y=0 only at some points
To solve for $y(x)$ in $y'=f(x)y$.
I was taught to consider case where $y=0$ and $y\neq0$.
For $y \neq 0$, can get $\int \frac{dy}{y} = \int{f(x) dx}$, then $y(x)=C e^{\int{f(x) dx}}$, where $C \neq 0$...
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Is finite-time explosion possible for a vector field $X:I\times \mathbb{R}\to \mathbb{R}^2$ of the form $X(s,t) = (1, f(s,t))$?
Let $I=(a,b)\subset \mathbb{R}$ be an open interval. Consider a vector field $X$ on $I\times \mathbb{R}$ of the form $$X(s,t)=(1,f(s,t)),$$
for some continuous function $f:I\times \mathbb{R}\to \...
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Intuition and Timescale Analysis for a Nonlinear Differential Equation Dependent on Parameter $K$
I am exploring the behavior of the following first-order nonlinear differential equation:
$$
\tau \frac{dy}{dt} = 2Ky^K(a - y^Kb)
$$
where $ y, a, b \in \mathbb{R} $, $ K \in \mathbb{N} $, and $\tau$ ...
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Solution to Stochastic Differential Equation Including Heaviside Step Function
There has been several entries on solving deterministic differential equations that include indicator functions. Stochastic differential equations may introduce new difficulties. Namely, does the ...
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Showing a function satisfying a certain differential inequality must always be positive
Suppose that
$$f''(t)+cf(t) \geq |f(t)|^p$$
for $t\geq0$, where $c>0$ and $p>1$, $f \in C^2([0, \infty))$, and we know also that $f(0)>0$ and $f'(0)>0$. Is it true that $f(t)\geq 0$ for ...
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Is there a source out there which works out a Hartman-Grobman-type theorem on a manifold?
My general goal is to understand more aspects of dynamical systems in the framework and language and differential geometry, and while for most things I can make out some sources which do that, I cant ...
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3
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56
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Why do we linearize about critical points?
I'm a beginner who's just learned about autonomous equations as an intro to non-linear diff. eqs. I've noticed that in the few sources I've used, on the topic of linearizing first order autonomous ...
- 133
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1
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Uniqueness of 1st order linear ODE with step discontinuities in driving function
Consider the following first order ODE initial value problem with a discontinuous driving function composed of step functions:
$$y'(t) + y(t) = u_1(t) - u_2(t)$$
$$y(0) = 0$$
$$u_a(t) \equiv
\begin{...
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Some confusion about the 245th page of Hamilton's The Ricci flow on surfaces
When I read the Hamilton's The Ricci flow on surfaces (237- 262 of this). I fail to get the (2) from (1) in picture below, where the $T$ of (1) is any nonnegative real number. In fact, I feel it is ...
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Uniform Convergence and ODE
The following problem appeared on a past exam at my institution:
Suppose that for each $n\in\mathbb{N}$, $u_n:\mathbb{R}\rightarrow\mathbb{R}$ is a differentiable function satisfying $$u_n'(x)=F(u_n(x)...
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Seeking General Solution for Nonlinear ODE Without Initial Conditions
I am working on solving the following nonlinear second-order ordinary differential equation (ODE):
\begin{equation}
f''(x) + f'(x) f''(x) + x f'(x) - f(x) = 0
\end{equation}
I am looking for a ...
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Optimal bidding and the value of the game
I migrated from this post, where the problem is a bidding game between two players on the sum of two fair die. Specifically, Player 1 rolls one die and sees the outcome and so does Player 2. They don'...
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Solve $ y'' + \lambda y^2 = 0$
Solve $ y'' + \lambda y^2 = 0$
Attempt 1 : if $\lambda =0 $ .then it's trivial to solve.
If $ \lambda <0 $ ,then $y '' \ge 0$
In particular when $y '' > 0 $ for some interval .
Let $ y''= e^{u(...
- 2,713
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Is it valid to replace the limit of a function inside the indifnite integration when computing limits?
The original problem tells me that
$f(x)$ is continuous on $[0, +\infty)$
$f(x)$ has a horizontal asymptote $y=b\ne 0$, i.e.,
$$
\lim_{x\to+\infty}f(x)=b \ne 0
$$
Then it asks me to choice one ...
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Stability of dynamical system where part of the state goes to zero
Consider the following dynamical system:
$$
\begin{aligned}
\dot{x}_1 &= -x_1 + g_1(x_1,x_2)\\
\dot{x}_2 &= g_2(x_2)
\end{aligned}
$$
where $x_1\in\mathbb{R}^n$, $x_2\in\mathbb{R}^m$, $g_2(x_2)...
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Checking if a polynomial is odd or even
I am reading a paper: 'Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions' by Zhan, Blower, Chen, Zhu.
On page 14 we have a differential equation (4.29) for ...
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Locally Lipschitz with respect to a variable uniformly to another implies Lipschitz for every compact subset
I was reading a proof of Picard-Lindelöf theorem and there is a step in the proof which needs the following proposition, which I have not been able to prove.
Let $$f:A\subset{\mathbb{R}^{n+1}}\to{\...
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Exponential difference equation [closed]
How would I solve the exponential difference equation
X(t+1) = A*X(t)^B,
with A and B being constants?
Thanks in advance.
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Proving a property on a system on nonlinear ODEs
I have the following system of first-order nonlinear ODEs
$$
\begin{align}
\Lambda' &= - Y \frac{2 \kappa \Lambda^2 \Omega }{\Xi} + \frac{\Lambda \Xi}{\Omega} \\
\Phi' &= \Phi \frac{4(\kappa ...
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Differential equation of perpendicular lines
Determine the differential equation of all straight lines who are at a unit distance from the origin.
If we assume the line to be of the form $ax+by+c=0$,then unit distance condition implies $|c|=\...
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How to solve the ODE $Q'=Q^2 +rQ$?
When I read a paper, seemly, the author state that
$$
Q(t) =\frac{-r e^{rt}}{e^{rt}-1}
\tag{1}
$$
solve
$$
Q'=Q^2 +rQ
\tag{2}
$$
where $r$ is a positive constant. About ODE I know little, I can ...
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How do I non-dimensionalise this model?
I have two predator-prey population models:
\begin{align*}
\frac{dN}{dt} &= rN \left(1 - \frac{N}{k}\right) - v(N)P \\
\frac{dP}{dt} &= cv(N)P - mP - qEP
\end{align*}
where v(N) is
\begin{...
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Show that the coefficient functions of a differential equation are uniquely determined by any basis for the solutions [closed]
Consider the differential equation
$L(y) \equiv y′′(x) + p_1(x)y′(x) + p_0(x)y(x) = 0 , x \in I$ (1.1)
where the coefficient functions $p_1(x) , p_0(x)$ are continuous on the open interval $I$. Let $\{...
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1
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Jack Hales 'Ordinary Differential Equations' as a first text on ODE's?
A few months ago I asked a question on here about comparing Arnold's text and Birkhoff/Rota's text, and the responses pushed me towards Arnold's. However after reading this question ... What is a good ...
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1
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Solution to heat equation on disk with boundary condition oscillating in time
I am trying to solve analytically the heat equation $\partial_t u = \Delta u$ on the unit disk $D_1\subset\mathbb{R}^2$ with Dirichlet boundary condition $e^{ikt}$. That is, I want a solution ...
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How should I find the equilibrium points and the general equation for the phase paths?
Find the equilibrium points and the general equation for the phase paths of $\ddot{x}+\cos(x)=0$. Obtain the equation of the phase path joining two adjacent saddles.
Above is the problem statement and ...
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Sufficient and necessary conditions for a solution of a certain system of ode's involving the cross product to be defined on the sphere $\mathbb{S}^2$
Consider the system of equations on $p,v: I\to \mathbb{R}^3$
\begin{align*}
p'&= -a v+b \cos(ct+d) (p\times v),\\
v'&= ap+ b\sin (ct+d) (p\times v),
\end{align*}
for constants $a,b,c,d\in \...
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any method to know if an IVP has a unique solution, no solution or infinite solution without solving it?
Assume the IVP
$$\frac{dy}{dx}=f(x,y), y(a)=b.$$
is there a way to know the nature of the solution without explicitly solving it?
what I tried if $f$ is Lipschitz, then it proves uniqueness.
But there ...
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0
answers
26
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Is the periodically driven Duffing's oscillator uniformly hyperbolic?
I am currently trying to figure out whether there is any physical measure (or SRB measure) for a periodically driven Duffing's bi-stable oscillator given by the following differential equations:
$$
\...
- 21
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votes
1
answer
291
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How to solve $f(x)=f(0)+\frac{x}{f'(x_{0})+\frac{x^{2}}{f''(x_{0})+\ddots}}$ for $f(x)$?
Question
How to solve
\begin{align*}
f\left( x \right) &= f_{0} + \underset{k = 1}{\overset{\infty}{\operatorname{K}}}\left[ \frac{x^{k}}{f^{\left( k \right)}\left( x_{0} \right)} \right] = f_{0} +...
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1
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Find the solution of the differential equation $z(z+y^2)dx + z(z+x^2)dy - xy(x+y)dz = 0$.
I have verified its integrability but since the equation is not homogeneous, I am stuck in arriving at the final solution. Also if the given differential equation is integrable, why can't we proceed ...
- 55
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0
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25
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finding the transfer equation of $\frac{H(s)}{Q_d(s)}$ in liquid control system with hydraulic integral controller.
On problem B-4-10
So my attempt is as follows:
from the liquid control system we have
$(q_d+q_i-q_o)dt=Cdh$ since $R=\frac{h}{q_0}$ and $R=0.5$ then $q_0=2h$.
By the equibilirium system on the lever ...
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Let $f \in C^\infty$, $f(0) = 0$ and $f(x) = \frac{1}{x} \int_{\frac{x}{2}}^{\frac{3x}{2}} f(y) dy$ $\overset{?}\implies$ $f(x)=cx$
Let $f \in C^\infty$, $f(0) = 0$ and
$f(x) = \frac{1}{x} \int_{\frac{x}{2}}^{\frac{3x}{2}} f(y) dy$
$\overset{?}\implies$ $f(x)=cx$
I started by multiplying $x$ to the lhs. This gives $f(x)x = \int_{\...
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votes
1
answer
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When exactly does a differential equation have no degree?
I am currently taking a Differential Equations class for my Computer Engineering program and we're still at the introduction of the basic concepts for DEs. I am genuinely confused as to whether what ...
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0
answers
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Assistance Needed on Invariant Curves in a $C^1$ Vector Field with Two Parameters
I've been exploring a problem for the past two months, and I would greatly appreciate your insights. I'm working with a $C^1$ vector field defined by the equation $\dot{\mathbf{x}} = f(\mathbf{x}; \...
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