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

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2
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1answer
472 views

Cancelling differentials

I'll start with an example. In physics, $x(t)$ represents the $x$-position of a particle, and $v(t)$ its ($x$-)velocity. To determine the total displacement of a particle on the interval $[a, b]$, we ...
14
votes
1answer
548 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
12
votes
4answers
2k views

Examples of nonlinear ordinary differential equations with elementary solutions.

I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
11
votes
4answers
362 views

How to prove that $\frac{d^n}{dx^n}(x^2-1)^n=0$ has $n$ real roots?

How do I prove that $$\frac{d^n}{dx^n}(x^2-1)^n=0$$ has $n$ real roots?
10
votes
6answers
625 views

When to write “$dx$” in differentiation

I'm taking differential equations right now, and the lack of fundamental knowledge in calculus is kicking my butt. In class, my professor has done several implicit differentiations. I realize that ...
8
votes
1answer
746 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
7
votes
2answers
368 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem. Let $f(x):\Omega\longrightarrow \mathbb R^n$ be a $C^2$ function where $\Omega$ is an open subset of ...
6
votes
1answer
214 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
6
votes
2answers
342 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
6
votes
3answers
234 views

Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely ...
5
votes
2answers
970 views

Cleaning up the Great Lakes - Differential equation systems

Problem Details The idea of the problem is to find out how long it would take to flush the Great Lakes of pollution. They're set up as a series of five tanks and you are given inflow rates of clean ...
5
votes
1answer
86 views

Decomposite a vector field into two parts

Let A be a region in $\mathbb R^3$, and suppose $ \vec {\mathbf F}$ is a smooth vector field on A. I was asked to show that I can write $\vec {\mathbf F}=\vec {\mathbf F_1}+\vec {\mathbf F_2}$, s.t. ...
4
votes
6answers
1k views

ODE introduction textbook

Unfortunately I have reached the maximum number of math classes I can take for my undergraduate degree. I still wish to study basic ODEs and basic number theory. What is a good textbook with an ...
3
votes
1answer
43 views

Uniqueness of solutions to linear recurrence relations

I understand that if I have a linear homogeneous recurrence relation of the form $q_n = c_1 q_{n-1} + c_2 q_{n-2} + \cdots + c_d q_{n-d}$, I can construct the characteristic polynomial $p(t) = t^d - ...
3
votes
1answer
4k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
3
votes
2answers
427 views

Advection Diffusion Equation on Semi-Infinite Domain

Regarding the BVP $$u_t(x,t) - v u_x(x,t)=k u_{xx}(x,t),\qquad x\geq0$$ with BC $u_x(0, t)=0$ for $t\geq 0$, and parameters $v,k>0$, I have some questions. Does an expression for the Green's ...
2
votes
2answers
466 views

How to solve this ODE?

For a certain problem, I am trying to solve the ODE $$\ddot{z}(t) - \omega^2 z(t) = f_0 \Big(e^{-i(\omega+\delta)t}+e^{-i(\omega-\delta)t}\Big)$$ where $\omega$ is a real and $\delta$ is close to ...
1
vote
2answers
106 views

Suggestion for a Lyapunov function

Consider the differential system $$ x'=x+y $$ $$y'=x-y+xy$$ What would be a Lyapunov function for this system at $(0,0)$? I have considered functions $V(x,y)=ax^{2n}+by^{2m}$ but none of ...
1
vote
1answer
2k views

Pursuit Curve. Dog Chases Rabbit. Calculus 4.

(a) In Example 1.21, assume that $a$ is less than $b$ (so that $k$ is less than $1$) and find $y$ as a function of $x$. How far does the rabbit run before the dog catches him? (b) Assume now that ...
0
votes
1answer
116 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
8
votes
2answers
212 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
8
votes
1answer
410 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...
8
votes
1answer
172 views

Solving the differential equation $\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$

How do we solve the differential equation $$\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$$? I tried substituting $v=yx$ but I do not seem to be getting anywhere.Putting $u=x-2y$ yielded nothing better. ...
7
votes
1answer
98 views

Solve $y' + \frac1y + \frac1x =0$ Differential Equation

Do you have any suggestions for how to sole this differential equation? $y'+\frac1x + \frac1y =0$ ? :) I tried solving this by changing variable in the form of $v=x^\alpha*y^\beta$ but it didn't ...
7
votes
1answer
133 views

Periodic solution of differential equation y′=f(y)

Let $f∈C^∞(ℝ^2,ℝ^2)$ and $∀x∈ℝ^2$ $f(kx)=k^2f(x)$ for $k∈ℝ$ Show that any periodic solution of $y′=f(y)$ is constant. My attempt : Let $\lambda \in \mathbb{R}$. Let $g$ a periodic solution ...
7
votes
1answer
6k views

General Solution of a Differential Equation using Green's Function

My father recently lent me an old textbook of his, called Mathematical Methods of Physics by Mathews and Walker. I am working on the following exercise. Consider the differential equation ...
7
votes
1answer
403 views

ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
7
votes
1answer
2k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
6
votes
2answers
131 views

Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as ...
6
votes
2answers
776 views

General solution for $y^{iv}+ 2y''+y=\cos x$

Here is another problem from Mathews and Walker that has given me some trouble. 1-18. Find the general solution of $y^{iv}+ 2y''+y=\cos x$. Note: Thanks, everyone, for clearing up the ...
5
votes
1answer
101 views

projectile motion (with height) complicated

When a child standing on a horizontal path throws a ball, it leaves her hand from a point that is 90cm vertically above the path. The ball clears a 4.5 m high wall that is 10.5 m away from ...
4
votes
0answers
89 views

Green's function in a moving frame for a constant heat source

I am looking for the Green's function of the problem in two dimensions $r =(x,z)$, \begin{equation} \nabla^2g + \frac{v}{D}\frac{\partial g}{\partial z} = -\delta (r-r_0) \end{equation} Which ...
4
votes
2answers
110 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
4
votes
1answer
321 views

Background for studying and understanding Stochastic differential equations

Assume I have back ground of the following knowledge based on the textbook as : ODE : ODE by Tenenbaum Baby probability : Ross 's baby probability Baby real anlysis : Bartle's introduction to real ...
4
votes
1answer
175 views

Some Scaling Estimate for Heat Kernel

NOTE. I have rewritten the question to summarize my current progress on this question. The bounty is for completing what I have done so far, or by offering a more elegant solution probably based on ...
4
votes
1answer
547 views

The characteristic polynomial of a recurrence relation.

If I have a linear homogeneous recurrence relation $$y_n=c_1y_{n-1}+\ldots+c_ky_{n-k},$$ I can get its characteristic equation, which is $$r^k=c_1r^{k-1}+\ldots+c_k.$$ In particular for ...
4
votes
2answers
3k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
4
votes
1answer
120 views

Special Differential Equation (continued-2)

May you help me out in solving inhomogeneous differential equation looking like[this is radial part of Schrodinger equation]: ...
4
votes
4answers
1k views

Help with solving differential Equation using Exact Equation method

I need to learn how to solve differential equations using either the Exact Equation Approach and or the Special Integrating Factor methods. Below is a differential Equation to solve. $(2xy^2 + \cos ...
3
votes
1answer
129 views

Energy Method to show uniqueness of solution of PDE

In my notes there is the following example about the energy method. $$u_{tt}(x, t)-u_{xxtt}(x, t)-u_{xx}(x, t)=0, 0<x<1, t>0 \\ u(x, 0)=0 \\ u_t(x, 0)=0 \\ u_x(0, t)=0 \\ u_x(1, t)=0$$ ...
3
votes
1answer
115 views

What does a standalone $dx$ mean?

Some literature uses $dx$, in the context of differential equations, in a confusing way without defining what it really stands for: $Mdx + Ndy = 0$ Does it mean one of the following or something ...
3
votes
3answers
155 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
3
votes
1answer
4k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
3
votes
0answers
154 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
3
votes
1answer
550 views

Explain the error term in Euler method

Task: I had to find out some estimates for M and L to make sure the proportional accucrazy is not above $10^{-4}$ in the Euler method with the problem below. I am trying to understand the page 672 on ...
3
votes
4answers
1k views

particular solution of $4y''-y= \sin(x)\cdot \cos(x/2)$

So I'm working with a nonhomogeneous second order differential equation: $$4y''-y=\sin(x)\cos(x/2).$$ I know that the general solution, $y$, equals $y_c + y_p$ where $y_c$ is the general ...
2
votes
1answer
89 views

Stability analysis for a system of two differential equations

I have this system of differential equations: \begin{equation} \frac{dx}{dt}=\alpha x-\beta xy\\ \frac{dy}{dt}=\beta xy-\gamma y \end{equation} I need to find the critical points and then do a ...
2
votes
1answer
129 views

Counter-example to Cauchy-Peano-Arzela theorem

I was looking for a counter-example to Cauchy-Peano-Arzela theorem. I found this paper (in french) from Dieudonné. [acta.fyx.hu] Take $E = c_0$ to be the space of real sequences converging to $0$, ...
2
votes
0answers
132 views

perturbation question

I'm a little stuck with a problem and I was hoping that you guys could help. Question: A projectile is fired up from the earth with an initial velocity of $v_0$ upwards. Accounting for air resistance, ...
2
votes
1answer
281 views

How to get the linear equation system for finite element method from the variational formulation

Let the problem be $$-u'' + a(x) u = f , \;x \in \Omega = ]0,1[ , \;u(0) = \alpha ; \;u(1) = \beta,$$ where $f \in L^2(\Omega) , a(x) \geq a_0 > 0 , a(x) \in L^{\infty}(\Omega).$ This problem ...