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

learn more… | top users | synonyms (1)

2
votes
0answers
65 views

List of ODE's that can be solved by Fourier transform

I am teaching introductory level Fourier analysis and I want to give my students some basic and some not so basic examples of how to solve ordinary differential equations with the method of Fourier ...
2
votes
0answers
42 views

Nondimensionization of a simple system.

A damped spring mass system is modelled below: $$m\frac{d^2y}{dt^2}=F_s+F_d\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space t>0$$ ...
2
votes
0answers
31 views

Find the initial movement of a particle

A particle with mass $m$ is moving along a curve and the force exerted on it always points towards the origin, and it´s magnitude is proportional to the distance between the particle and the origin, ...
2
votes
1answer
96 views

Proving entire function is constant

Let $f(z),z^5\bar{f}(z)$ be entire functions on $\mathbb{C}$. Show that $f$ is constant. I tried using Cauchy-Riemann quations in their polar form in order to find out the derivaties are zero and ...
2
votes
0answers
60 views

Period of a pendulum

Consider the pendulum problem $\frac{d^2x}{dt^2}+\sin(x)=0$ $\frac{dx}{dt}(0)=v_0=0$ $x(0)=x_0$ Show that the period ...
2
votes
0answers
32 views

2nd order linear differential equation with non-constant coefficients

Considering the equation $2y''+(x+1)y'+3y$ where $X_0=2$. Find the general term in each solution. That is, the general term for Y1,Y2 where $y=A_0(Y_1)+A_1(Y_2)$ I've solved this as ...
2
votes
0answers
90 views

Validity of approximating a difference equation with a differential equation

Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle ...
2
votes
0answers
51 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
2
votes
0answers
30 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
2
votes
0answers
30 views

Solve the following ODE

Solve the following ODE $$(y-x)\left(1+x^2 \right)^{\frac{1}{2}}\dfrac{\mathrm{d}y}{\mathrm{d}x}=n\left(1+y^2 \right)^{\frac{3}{2}}$$ I have tried substituting $y=\tan \theta$ and $x=\tan \phi$ ...
2
votes
0answers
54 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
2
votes
0answers
52 views

List of eigenvalues for the Schrödinger equation

I'm writing an algorithm which computes the eigenvalues $E$ of the Schrödinger equation with potential $V(x) = x^2$, ie the harmonic oscillator. The equation is defined as follows $$ y''(x) = ...
2
votes
0answers
38 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
2
votes
1answer
85 views

Hints on solution to $u_t-\Delta u+cu=f$

Consider the problem (Evans, Ch 2, 14) $$ u_t-\Delta u+cu=f ,x \in \mathbb R^n\times (0,\infty)$$ $$ u=g , \mathbb R^n\times {t=0} $$ If $u$ solves $ u_t-\Delta u=f$, $u=0$ on and $v$ solves ...
2
votes
1answer
56 views

Is it possible to separate two variables?

Here is the following problem: $$ (\frac{du}{dv})^2=u(v-u)$$ Is it possible to separate these two variables?
2
votes
0answers
47 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
2
votes
1answer
143 views

What could be the mathematical model behind “beginner's luck” (followed by losses) in gambling?

I recall a documentary in which a slot machine had trial runs and at first, the desired "bingo" outcome came out more often, but later waned into losses. A scientist plotted the graph, a discrete ...
2
votes
0answers
45 views

How can I prove these two fields are locally topologically conjugated?

The problem is to prove that the fields $x'=x$ and $x'=x^3$ are locally topologically conjugated in the origin. I found that the corresponding flux for the first equation is $\phi(x_0,t)=x_0e^t$.The ...
2
votes
1answer
42 views

Periodic nature of ODEs

Generally speaking without solving, how are periodic function solutions a priori recognized by any inspection or computation of terms contained in their ODEs? (Prior knowledge is a not a ...
2
votes
0answers
240 views

Math software for plotting phase portraits

I'm looking for math software which is possible to plot phase portraits for ODE and systems of differential equations. Is there a software which can create not only simple 2D phase portrait plots but ...
2
votes
1answer
152 views

Simple Example of Method of Characteristics

To find the general solution of an equation such as $u_{x} - yu_{y} = 0$, it is clear by the method of characteristics that the characteristic curves satisfy $dx = -\frac{dy}{y}$ and so we get the ...
2
votes
0answers
44 views

Nontrivial solutions for a system of equations

Consider $t:[0,1]^2\to R$ that is differentiable a.e. and satisfies conditions (i)-(ii): (i) $$ \int_0^1 \frac{\partial t}{\partial t_1}(x,y)f(y\mid x)\,dy=0, \quad \forall x\in[0,1] \\ \int_0^1 ...
2
votes
0answers
64 views

How to classify the equation $\frac{dy}{dx} + x^{2}y = xe^x$

I have the following "homework" problem: Classify each of the following differential equations as ordinary or partial differential equations; state the order of the equation; and determine ...
2
votes
0answers
76 views

difference between runge kutta methods of same order

I recently read about runge kutta methods for solving differential equations. So far I understood the idea but up to know nobody could answer me following question: If we consider the explicit rk ...
2
votes
0answers
87 views

Understanding the eigenvalue problem: $x^2y''+xy'+\lambda y = 0$

I would just like to clarify a fw things I am not really understanding about Sturm-Liouville forms and eigenvalue problems: I have the practice question: $x^2y''+xy'+\lambda y = 0$ with boundary ...
2
votes
1answer
86 views

Idea to solve $\frac{dy}{dx}+x*(x+y)=x^3*(x+y)^3$ ?

I find this equation in this website http://www.prise2tete.fr/forum/viewtopic.php?id=10003 $\frac{dy}{dx}+x*(x+y)=x^3*(x+y)^3$ But the author don't answer and i want to know if it is possible to ...
2
votes
0answers
77 views

Green's function for Dirichlet Laplacian

I am thinking of the Dirichlet boundary condition $u|_{\partial \Omega}$ for a domain $\Omega \subset \mathbb{R}^n$. Let $\Delta$ be the Dirichlet Laplacian, which accepts only functions with the ...
2
votes
2answers
81 views

General form for finding tangent that intersects a point not on the curve

Particular cases of this problem have previously been addressed here and here, but I'm interested in the general case of the following problem: Given a function $f(x)$ and a point $P = (x_0, ...
2
votes
0answers
54 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
2
votes
0answers
24 views

Existenence of the solution for a PDE-ODE system.

I have the PDE-ODE system below: $\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$ $\frac{dv}{dt}=-\alpha cv+\xi(c,v)$ with initial conditions and Neumann boundary ...
2
votes
0answers
43 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
2
votes
0answers
104 views

How to solve second order differential equations? [summary]

As I do my engineering studies, I find more and more ways to solve differential equations, especially the second order ones. With more and more ways to solve these equations, I am loosing my overview ...
2
votes
1answer
33 views

factor $y'$ as $y' = f(x)g(y)$

I have a task where I have to write following differential equation as $y' = f(x)g(y)$ but I see no way you could factor it into two functions each only depending on $x$ respectively $y$: $$y' = ...
2
votes
0answers
28 views

How do I approximate $f''(x)+(E-U(x))f(x)=0$ for a piecewise $U$ and find $E$?

I am trying to approximate the solution to the equation $f''(x)+(E-U(x))f(x)=0$ where $U(x) = \begin{cases} \frac{U_0}{m}x-U_0 & \text{for $-m<x<0$} \\ \frac{-U_0}{m}x-U_0 & ...
2
votes
1answer
52 views

Example of cyclic vectors in linear differential equations

Suppose $f(x)$ obeys the first-order differential equation $f'(x) = P(x) f(x)$, and $g(x)$ obeys the first order differential equation $g'(x) = Q(x)g(x)$. Is there a second-order differential ...
2
votes
1answer
51 views

nonlinear, nonhomogeneous ODE 1. order

Solve $x'(t)-\dfrac{a}{t}x(t)=b(t),~a=const,~x(0)=0$. Homogeneous Solution: $\dfrac{x'(t)}{x(t)}=\dfrac{a}{t}\quad|\int\\ \ln(x(t))=a\ln(t)+c,~c=const\quad| e\\ x(t)=t^ae^c$ Is that correct? No ...
2
votes
0answers
40 views

How to solve ordinary differential inequations with vector variables?

Given $a\in\mathcal{R}_+^d$ and $s\in\mathcal{R}^d$,we wanna a function f(.) which maps s to a vector $f=\begin{bmatrix}f(s_1),\cdots,f(s_d)\end{bmatrix}^T$ and satisify the following inequation. ...
2
votes
1answer
79 views

rough question in Differential Equation.

I'm trying to solve the following system of differential equations, but I couldn't find any method / procedure to obtain the solution. I don't want a comprehensive and complete answer; a hint will ...
2
votes
1answer
100 views

Periodic solution to differential equation [closed]

For each $\epsilon>0$, show that the differential equation $$x'=x^2-1-\cos(t)-\epsilon$$ has at least one periodic solution with $0<x(t)<\sqrt{2+\epsilon}$.
2
votes
0answers
41 views

Solution of $\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$

How do we solve $$\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$$ I suspect it will be a function of many cases. The solution of $$\Pi(x+1)+\sin(x)=y(x)+y'(x)$$ is hard only at the evaluation of the last integral ...
2
votes
0answers
44 views

Laplace transform for solving differential equation ? help?

I have a differential equation: $$y'' + 4y' + 3y = 6t + 14, $$ with initial conditions $y(1) = 1,~y'(1) = 0.5$. Can I use the Laplace transform to solve this equation whose initial condition is not ...
2
votes
0answers
37 views

The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
2
votes
0answers
107 views

Forced oscillation in a pendulum and resonances

In a pendulum without the small angles approximation the equation describing the motion of the mass is: $$\ddot{\phi}(t)=-\dfrac{g}{l}\sin\left(\phi(t)\right)$$ Applying a sinusoidal force ...
2
votes
0answers
55 views

Does the IVP $y'=1+y^{2/3}$ with $y(0)=0$ have a unique solution?

Consider the IVP $y'=1+y^{\frac{2}{3}}$ with $y(0)=0$, I've to show it has unique solution I can't apply Picard's theroem here since $f(x,y)=1+y^{\frac{2}{3}}$ is not Lipschitz , but I can apply ...
2
votes
0answers
60 views

Taking partial derivatives over multiple summations

I have the following equation obtained from one of the models. $\mathcal{H} = \sum\limits_{D} \sum\limits_{W}n(d,w)\sum\limits_{Z} p(z|d,w)[\log{p(d)}+\log{p(z|d)}+\log{p(w|z)]}$ I need to take ...
2
votes
0answers
26 views

A problem with Riccati's equation

Solve $$ x'=-\frac {4}{t^2}-\frac 1 t x+x^2$$ knowing that $\gamma (t)=\frac 2 t $ is a particular solution. So I make a substitution $x=\gamma (t)+\frac 1 u$ $$x=\frac 2 t +\frac 1 u$$ ...
2
votes
0answers
33 views

Catagorising a Differential Equation

I have $$ ...
2
votes
0answers
46 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
2
votes
1answer
282 views

When is the solution to a n initial value problem matrix differential equation invertible?

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
2
votes
0answers
46 views

Is ODE theory useful for developing numerical solvers for ODEs?

I will be doing research in developing numerical solvers for ODEs. I was wondering if knowledge of ODE theory will be useful and if so in what ways. I am asking because, I am inclined to take a ...