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.

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what is the general solution to the ODE [closed]

What is the general solution to the ODE I would like to find the general solution to the ODE: -2y' + y = 0
luffy's user avatar
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About ODE $2p'q-3pq'=\lambda$

I need to give any possible characterization (could be geometric, algebraic or of other type) of a pair of functions $p$ and $q$, holomorphic over $\mathbb{C}$, and a complex number $\lambda$ ...
MFS_math's user avatar
1 vote
1 answer
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Jacobian of ODE in polar coordinates

The system of ODEs $$\dot{u} = bu - v + au(u^2 + v^2)$$ $$\dot{v} = u + bv + av(u^2 + v^2)$$ can be written in polar coordinates as $$\dot{r} = br + ar^3$$ $$\dot{\phi} = 1$$ I know that in Euclidean ...
Sugario's user avatar
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Continuous vs. Annual compound interest

A saving amount pays 8% interest per year compounded continuosly. In addition, income from another investment is credited to the amount continuosly at the rate $400 per year. Express this physical ...
Vinay5101's user avatar
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Can A General Solution Be Obtained If A Particular Solution Is Known For A System of Differential Equations?

This question occured to me while trying to solve a three-body-esque problem involving three charged particles placed along a straight line (ignoring effects of electromagnetic radiation etc. etc.). ...
davidaddisonsenjaya's user avatar
2 votes
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When a sequence satisfies $a_n= \sum_{i=1}^{n-1} f_i(n) a_i+ g(n)$?

Question Given a sequence $a_n$ when it is possible to express such sequence as a "linear recurrence relation with not constant coefficients"? i.e. when $$ a_n= \sum_{i=1}^{n-1} f_i(n) a_{i} ...
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Is there analytic eigenvalue and eigenfunction of the anharmonic oscillator?

Both the analytic eigenvalue and eigenfunction of the harmonic oscillator $\frac{d^2 y(x)}{dx^2}-x^2 y(x)=\lambda y(x)$ are known. Is there an analytic expression for the eigenvalue and eigenfunction ...
ssskkkky's user avatar
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1 answer
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How is it possible to demonstrate the necessary system to find particular solutions in LDE of order 2

When when want to find particular solutions to a LDE of order 2 like : $$ y'' + ay' + by = f(x) $$ I see in a lot of source that says that after finding two solutions for the homogeneous equations $...
jozinho22's user avatar
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6 votes
2 answers
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In what time $\tau$ will the particle reach the point $x=0$?

A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its ...
Andrés de Fonollosa's user avatar
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Finding parameter values for which a system has closed orbits

My question is on Exercise 7.3.8 of Chaos and Nonlinear Dynamics (2nd ed) by Strogatz: 7.3.8. Recall the system $\dot{r} = r(1-r^2) + \mu r \cos \theta, \; \dot{\theta} = 1$ of Example 7.3.1. Using ...
Leonidas's user avatar
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Checking if a first order differential equation follows certain initial conditions

I am trying to integrate the following first-order ordinary differential equation and confirming whether it is possible such that $t(r) = 0$ when $r = 0$. $$\int \frac{dr}{r^2((a - r)^2 + b)} = \int c ...
Arjo's user avatar
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1 answer
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Proving solutions of $y''+p(x)y'+q(x)y=0$ to be linearly independent

When studying Elementary Differential Equations by William, I found trouble understanding Theorem 5.1.5 It says the two solutions are linearly independent iff their Wronskian is never zero, but I ...
AntidusPig's user avatar
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Difference between finite difference approximation and differential quadrature approximation

As a student of numerical analysis, I understand that a finite difference approximation (FDM) of the derivative '$u_x$' of a desired solution '$u$' at the point $x_n$ in the domain is just a linear ...
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Numerical Analysis of Differential Equation with Boundary Conditions

Note I have decided to edit this post so it is more specific and also because I was incorrect the first time. I didn't wanna make another post about the same subject, but if this is not allowed then ...
Need_MathHelp's user avatar
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find time needed for a given displacement where velocity is nonlinear in time

A velocity v is obtained as a function of time t from a nonlinear ordinary differential equation and thus the inverse function t(v) can't be written down analytically in general. How to find out the ...
feynman's user avatar
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Necessary condition for exponential stability of an LTV system

This is a homework problem from my adaptive control course: Consider the IVP $\dot x(t) = -u(t)^2x(t)$ with $x(0) = x_0$. Suppose the system is exponentially stable. Show that there exist some $\...
ArGenya's user avatar
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what more can I say using Sturm's comparison theorem?

Can I use Sturm's comparison theorem to say something about the average? Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function $$ \...
alejandro's user avatar
1 vote
1 answer
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How to derive an inequality for the solution of a differential equation based on a differential inequality?

Suppose that the following differential inequality holds: $$ \dot{y}(t) \leq -c\cdot y(t) + \lambda \ \ (1)$$ with $c,\lambda > 0 $ positive constants. Now let $\rho = \lambda /c > 0$, I'm ...
Teo Protoulis's user avatar
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Proof of ellipticity of lemniscate functions from integral definition

The lemniscate functions $\text{sl}$ and $\text{cl}$ are the solutions to the differential equation $$ (y')^2+y^4=1$$ with $\ y(0)=0, \ y'(0)=1$ $\ $ or $\ $ $y(0)=1, \ y'(0)=0.$ Using the integral ...
Noa Arvidsson's user avatar
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102 views
+50

Gradient estimates of linear elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ $$u(x)\Big|_{\...
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What does it mean by asking a solution to a given function?

Question I have a question like this, and my question is that what does it mean by asking how many different solutions? It is already given as a piecewise. Is it a solution to say if a<c, b=1? Or ...
Ege's user avatar
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Sum to integral for differential equation [closed]

I read a proof and i don’t understand this : $\sum_{i=0}^{x}[ f(x-i)-2f(x-i-1)+f(x-i-2)]=D(x)-D(x-1)$ which implies that $D’(x)=\int_{0}^{x}f’’(x-s)ds$ Can you help me? Thanks
leanig23's user avatar
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Laplace Transform of a Piece-wise function with a Weibull distribution.

Suppose I have the following piecewise function: $$Q(t) = \begin{cases} W(t) & t<T \\ 1 & t=T \\ 0 & t>T \end{cases}$$ ...
Keyvan's user avatar
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Solving a first order differential equation - does it matter where you put the absolute value sign?

Problem: Solve the following differential equation: $$ 6x^2y \, dx - (x^3+1) \, dy = 0 $$ Answer: \begin{align*} 6x^2 \, dx - \dfrac{ (x^3+1)}{y} \, dy &= 0 \\ \dfrac{ 6x^2}{x^3+1} \, dx - \...
Bob's user avatar
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1 vote
1 answer
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Linearization of a nonlinear third order ODE and stability

I would like to know if the following differential equation ($\alpha,\beta,\gamma,d,\Lambda,w$ are constants) $x'''(t)=\frac{1}{24 (3 \alpha -\beta )}\frac{x(t)^{-3 w-2}}{x'(t)} \left(36 \alpha x(t)^{...
Axionlike particles's user avatar
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Uniqueness of a solution of a differential equation

Let $f,g \in L^{\infty}([0,1],\mathbb{R}^d$ and $F:\mathbb{R}^n \times \mathbb{R}^d \to \mathbb{R}^n$ be a $C^1$ function. Let us consider the two differential equations: $$ \dot{x}(t)=F(x(t),f(t)), \...
hanava331's user avatar
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0 answers
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Is there a theory of "quadratic" Hamiltonian evolutions on Poisson manifolds?

I am dealing with a PDE which can be written in the form $$\frac{d}{dt} f(t) = \{a, f(t)\} + \{\{b, f(t)\}, f(t)\}$$ A Hamiltonian equation on a Poisson manifold has the following form: $$\frac{d}{dt} ...
Robert Wegner's user avatar
-2 votes
1 answer
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I want to know the behavior of the solution without solving the ordinary differential equation for enzyme-substrate reactions.

I want to know the behavior of the solution without solving the ordinary differential equation for the following enzyme-substrate reactions. Here, $k_1,k_2,k_3>0$, and [S] is the concentration of ...
Blue Various's user avatar
1 vote
1 answer
72 views

Initial conditions to obtain a steady solution when an ordinary differential equation is given

In a chemical reaction that follows the logistic equation, if the initial velocity is 0, it becomes a steady solution. However, in the equation of motion, the fact that the initial velocity is 0 does ...
Blue Various's user avatar
3 votes
1 answer
107 views

Find an asymptotic solution of a ODE system

Consider the following ODE system: $\frac{dx}{dt}=x^2+y^2-y\\ \frac{dy}{dt}=-2xy-x$ I want to prove that there exists a solution $(x(t),y(t))$ such that $\lim_{t\to +\infty}(x(t),y(t))=\lim_{t\to -\...
Taylor Yao's user avatar
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0 answers
26 views

Understanding the Euler operator with weights [closed]

How to solve $$\sum_{i=1}^n a_j x_j D_j f + c^2 f=0 $$
user1289267's user avatar
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0 answers
41 views

How to test if a certain differential equation has an analytic solution (solution has Taylor expansion)?

How to test if the differential equation $$(1+x)y'''+y y''-y'^2+(2+x)y''+y'=0,$$ $$y(0)=1, y'(0)=1, y'(\infty)=0$$ Has an analytic solution? Solution that equals its Taylor series?
Mohamed Mostafa's user avatar
-2 votes
1 answer
64 views

Global solution of $y'=y^4-x^8$ [closed]

I encountered a problem while working on a mathematical analysis exercise. The problem is as follows: we need to determine the initial value condition $y(x_0)=y_0$ for the ordinary differential ...
Liping Li's user avatar
2 votes
0 answers
48 views

Technique for generating Lie point symmetries

Consider I believe that there is something wrong with this text. In particular, how is $$\Delta=0 \quad \Longrightarrow \quad V(\Delta)=0$$ completely non trivial by linearity of operators? Moreover, ...
Maths Wizzard's user avatar
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How to prove two functions cannot be solutions of a same ODE?

Question: Consider here the two functions $f: \mathbb R \rightarrow \mathbb R$ and $g: \mathbb R \rightarrow \mathbb R$ defined by $f(t=-t^2 +t + 1$ and $g(t)= cos(t)+sin(t)$. Show that $f, g$ are not ...
chen zhang's user avatar
1 vote
1 answer
51 views

Equivalence between IVP and integral equation

I was trying to prove the following theorem: If $f(x, y)$ is continuous on some region $R \subseteq \mathbb R^2$ then any solution of IVP $$y'(x)=f(x,y(x)), y(x_0)=y_0 \tag{1}$$ is also a solution of ...
baja1997's user avatar
1 vote
0 answers
72 views

An extremely rigorous and formal definition of differential equation.

I actually asked a version of this question before, here: What is the formal, rigorous definition of a differential equation?. However, I should have asked what an equation is, first, which I did here:...
user107952's user avatar
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Determining if the pde ($[w(x)u_x]_x -v(x)u_{tt} =0$) can be reduced to an ode using seperation of variables.

So far I have: $u(x,t)=f(x)g(t)$, $u_x(x,t)=f'(x)g(t)$, $u_{xx}(x,t)=f''(x)g(t)$, $u_t(x,t)=f(x)g'(t)$, $u_{tt}(x,t)=f(x)g''(t)$ $w(x)f''(x)g(t)-v(x)f(x)g''(t)=0$ $\implies w(x)f''(x)g(t)=v(x)f(x)g''(...
PeakyBlaze7788's user avatar
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0 answers
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Relation between PDE and ODE [closed]

It was mentioned somewhere [A very good source i.e. in a Book] that Partial Differential Equations are essentially infinite number of Ordinary Differential Equations. I couldn't reconcile this ...
Dan's user avatar
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1 vote
2 answers
42 views

Finding integrating factor for a for some differential

I have the following problem : Find the integration factor $\mu=\mu(x,y)$ for the PDE : $$(2x^3y-y^2)dx - (2x^4 + xy)dy = 0 $$ I tried to solve it but I was unable to find the solution. If possible, I ...
arofenitra's user avatar
1 vote
0 answers
22 views

Does the following Young differential equation with a functional locally Lipschitz derivative have a finite solution?

I wanted to prove the following statement: for $F \colon \mathbb{R}^d \to \mathbb{R}^d$ and $W \colon [0,1] \to \mathbb{R}^d$, we have the following Young differential equation $$ dH_t = F(H_t) dW_t, \...
user807606's user avatar
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0 answers
17 views

Can power series method fails to solve differential equation at ordinary point?

Is there a sufficient condition if satisfied by a differential equation at an ordinary point ensures finding the solution using power series method? In other words, can power series method fails to ...
Mohamed Mostafa's user avatar
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0 answers
16 views

Is state variable $S$ is continuous as parameter vary for a system of differential equation?

Given, simple model, $$\begin{eqnarray} \frac{\mathrm{d}S}{\mathrm{d}t} &=& -\beta SI \\ \nonumber \frac{\mathrm{d}I}{\mathrm{d}t} &=& \beta SI - \gamma I \\ \nonumber \frac{\...
A learner's user avatar
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2 votes
2 answers
103 views

I need an example of a function such that the derivative minus x is the same as the original function [closed]

I'm curious if there is a function F(x) whose derivative f(x) is the same as F(x) + x. I tried to make one myself by adding things to e^x with no success. the function was mentioned in Question 21 of ...
신규민's user avatar
0 votes
1 answer
47 views

Solution of an IVP through Laplace transform

Let $𝑦(𝑡)$ be the solution of the initial value problem $$y''+4y=\begin{cases} t, & 0\leq t\leq 2\\ 2, & 2<t<\infty \end{cases}.$$ Also, it is given that $$y(0)=0, y'(0)=0.$$ Given ...
PAMG's user avatar
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0 answers
20 views

Non linear population model analysis

I started watching a course on ODE, In the course lecture first video, the teacher gave some motivations, this example of Non-Linear population model is first among them, the equation is given by $$\...
Praveen Kumaran P's user avatar
-4 votes
0 answers
45 views

Differential equation population growth [closed]

The population of a certain town is directly proportional to the square root of the present population at any time. If the population initially is 20000 how much the population after 10 years?
Kamran Khan's user avatar
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0 answers
16 views

Legendre functions of the second kind with negative integer degree

I have been recently reading about properties of Legendre functions in several sources and cannot seem to find any properties of Legendre functions of the second kind with negative integer degree. For ...
Lawford Hatcher's user avatar
4 votes
1 answer
32 views

Question about properties of Picard-Lindeloef existence theorem

I have a few questions about solutions that arise from differential equations where Picard-Lindeloef can be applied: In the problem $y'=f(t,y)=-y^2$, solutions have the form $\frac{1}{x-c}$ and ...
John Doe's user avatar
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0 answers
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Solving a funky differential equation.

I'm currently trying to solve the DE that defines charge in a circuit containing an Inductor, Capacitor, Resistor and (crucially) a Memristor. This needs to be able to work for any variable values and ...
Seb's user avatar
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