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

learn more… | top users | synonyms (1)

8
votes
1answer
209 views

What's the name of this chaotic system? (Cool pics included.)

I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a ...
8
votes
1answer
929 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 ...
8
votes
3answers
582 views

Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer?

This is a slightly farcical question, for which I apologise. An undergraduate tutee of mine was faced with the following problem: Q. A particle of mass $m$ moving along a line is subject to a force ...
8
votes
3answers
1k views

Variation of Parameters Differential Equations Derivation

So I never fully understood the derivation of the method of variation of parameters. Consider the simplest case $$y'' + p(x)y' + q(x)y = f(x)$$ And the homogenous solutions is $y_h=c_1y_1+c_2y_2$ ...
8
votes
3answers
173 views

How to get the solution to these differential equations

I would like to get from $$ \tan(x) = \frac{y''}{y'} + y' $$ The answer is $$ y = \ln(c_1\tanh^{-1}(\tan(\frac{x}{2}))+c_2) $$ The other equation is $$ \sec(x) = \frac{y''}{y'}+y' $$ The answer ...
8
votes
1answer
228 views

When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
8
votes
1answer
117 views

$|\nabla f (x)| =1$ implies $f$ linear?

Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is smooth and $|\nabla f (x)| = 1$. Must $f$ be linear (up to an additive constant)? That is, must we have $f(x) = a\cdot x +b$ for constant ...
8
votes
1answer
196 views

$x''= \frac{Ax+B}{Cx+D}$

Might there be a closed-form solution to the second-order differential equation below?$$x''(t)=\frac{Ax+B}{Cx+D}$$ If not, is there any way to get a power series approximation in terms of the ...
8
votes
4answers
1k views

What comes after Differential Equations?

First of all, please do excuse the lack of correct terminology, I've haven't learnt Differential Equations at school (yet) so this question comes from just a bit of research I did for my own ...
8
votes
1answer
3k views

A Differential Equation with Trigonometric Coefficients

Suppose we have the following second-order differential equation: $\cos^2(x)y'' -\sin(x)y' + y = 0$ How do we determine its general solution? I couldn't even guess a particular solution; all my ...
8
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 ...
8
votes
1answer
99 views

An MCQ on Greens function

$$G(x,t) =\begin{cases} a+ b\log t & \text{if $0<x<t$ } \\[2ex] c+ d\log t & \text{if $t<x<1$ } \end{cases}$$ is a Greens function for $xy''+y'=0$ subject to $y$ being bounded as ...
8
votes
1answer
132 views

How to estimate solutions to an ODE with an asymptotically nilpotent coefficient?

Suppose $f:\mathbb R\to\mathbb R^n$ satisfies $$ f'(t) = A(t)f(t), $$ where $A$ is a smooth matrix-valued function. If I know that the matrix $A(t)$ is asymptotically nilpotent, how could I prove a ...
8
votes
1answer
94 views

Stochastic Calculus Question

I'm new here and was hoping someone could help me answer this question. I'm reading a paper and I'm a bit confused on how they go from 1 equation to the next. They say: Let \begin{align} x(t) = {} ...
8
votes
1answer
305 views

First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no ...
8
votes
1answer
282 views

Solving a differential equation

OK, so, I'm supposed to solve the differential equation $$\frac{dy}{dx} = \frac{y+2x}{y-2x}$$ by making the substitution $y = ux$, to make the equation separable. Then $$\frac{dy}{dx} = u + ...
8
votes
1answer
514 views

Grothendieck connections and jets

The following question is based on some remarks in section I.2 of Deligne's book Equations Différentielles à Points Singuliers Réguliers. Let $X$ be a smooth complex variety and ...
8
votes
1answer
77 views

Solving the differential equation $x x''(t)=\frac{1}{t^3-t}$

Solve the following differential equation $$x x''(t)=\frac{1}{t^3-t}$$ I tried to integrate both members: $$x(t) x'(t)-\int [x'(t)]^2 dt=\frac{1}{t}+\log\left|\frac{1}{t}-1\right|$$ but the situation ...
8
votes
2answers
1k views

Frobenius method, why is it an issue when the roots of the indicial equation differ by an integer

When solving second-order differential equations by the Frobenius method at a regular singular point, you are supposed to use the two roots of the indicial equation to give you two independent ...
8
votes
3answers
594 views

ODE with singular coefficients

I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - q(x) y(x) = \lambda y(x).$ Then I got a more sophisticated differential equation ( second one) and is given by $$-(1-x^2)y''(x) +x ...
8
votes
1answer
122 views

What condition ensures the solution is periodic? (ODE)

Suppose that $\phi$ is a solution to the ODE \begin{align} x' = f(x) + \sin(t) \end{align} What condition can we put on $f$ to ensure that $\phi$ is periodic of period $2\pi$? That is, what do we ...
8
votes
1answer
172 views

An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
8
votes
3answers
345 views

Why can't I solve this homogenous second order differential equation?

I've been banging my head on the wall for quite some time trying to come up with a solution to the following: $$\frac {\partial^2 y(x)} {\partial x^2} + (A-B*V(x)) y(x) = 0 $$ $$V(x) = (36 + (2 - ...
8
votes
0answers
234 views

Uniqueness of solutions to $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$

The problem I am working on is to show that there is a unique compactly supported solution to the PDE $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$, $(x, t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0)= ...
7
votes
6answers
261 views

Are the any **non-trivial** functions where $f(x)=f'(x)$ not of the form $Ae^x$

This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
7
votes
5answers
469 views

Solving a separable differential equation

Solve the differential equation: $$y'=\frac{1-y^2}{1-x^2}$$ My book says the solution is: $$y=\frac{x+c}{cx+1},$$ where $c$ is a constant. It's been ten minutes I tried to verify if it was correct ...
7
votes
4answers
1k views

How to solve $y''' - y = 2\sin(x)$

$$y''' - y = 2\sin(x)$$ I'm doing differential equations and know pretty much all methods of solving them, but I haven't come across anything of a higher order than second yet. How do I go about ...
7
votes
3answers
1k views

Solving $-u''(x) = \delta(x)$

A question asks us to solve the differential equation $-u''(x) = \delta(x)$ with boundary conditions $u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside ...
7
votes
3answers
302 views

Find solutions of the differential equation $3x^2y''+5xy'+3xy=0$.

Find all the solutions of the form $y(x)= x^m \sum_{n=0}^{\infty} a_nx^n, \ x>0 (m \in \mathbb{R})$ of the differential equation $3x^2y''+5xy'+3xy=0$. That's what I have tried: Since $x>0$ the ...
7
votes
3answers
577 views

Common term for differential equations and recurrence relations

Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you ...
7
votes
3answers
331 views

Why it is absolutely mistaken to cancel out differentials?

In many of my physics courses, (don't worry, this is a mathematics question!) My teachers cancel out differentials, and every time, they say: "If a mathematician saw me canceling out this ...
7
votes
2answers
860 views

Manifold interpretation of Navier-Stokes equations

I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines ...
7
votes
5answers
2k views

Modelling forces acting on a sail

I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship) A couple of things I was thinking about: 1) Can create a very simple model where wind is 'one' force ...
7
votes
3answers
423 views

Finding matrix exponential

I am trying to compute the matrix exponential for $$A=\left( \begin{array}{ccc} 1 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & -1 ...
7
votes
2answers
415 views

solving the integral $ \int_0^\infty \dfrac{\sin xt}{x(x^2+1)} dx $

How to solve the following integral by differential equations techniques? $$ \int_0^\infty \dfrac{\sin xt}{x(x^2+1)} dx $$
7
votes
3answers
968 views

Square root of differential operator

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$. How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number) For ...
7
votes
3answers
358 views

Counterintuitive PDE

After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following: Given the PDE $\dfrac{\partial f}{\partial x} = 0 \quad $ on $U ...
7
votes
3answers
1k views

Solve $\frac{dx}{dt} = x^3 + x$ for $x$

This is a seemingly simple first order separable differential equation that I'm getting stuck on. This is what I have so far: $$\frac{dx}{dt} = x^3+x$$ goes to $$\frac{dx}{x(1+x^2)} = dt$$ Now ...
7
votes
1answer
1k views

How can I show that $y'=\sqrt{|y|}$ has infinitely many solutions?

Show that the first order differential equation $y'(x)=\sqrt{|y(x)|}$ with intial value $y(1/2)= 1/16$ has infinitely many solutions on the interval [−1, 1]. My thought were to show that this ...
7
votes
3answers
2k views

Why does the absolute value disappear when taking $e^{\ln|x|}$

I have noticed that if you have an equation (after integrating) such as $$\ln|y| = \ln|x| + c,$$ and you further simplify it using the law of exponents, you get $$e^{\ln|y|} = e^{\ln|x|+с},$$ which is ...
7
votes
2answers
260 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
7
votes
1answer
236 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $$f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$$ satisfies $$x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R.$$ How ...
7
votes
1answer
291 views

Finite dimensional spaces

What are the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$, then $\frac{df}{dx}$ is in $W$.
7
votes
4answers
447 views

Show $f''+vf' +\alpha^2 f(1-f)=0$ has solutions satisfying $\lim_{x \to - \infty}f=0$ and $\lim_{x \to \infty}f=1$ given $v\leq -2\alpha < 0$

I posted this question before but I took a completely different approach here, that's why I reposted as my previous question was already very long and took a different approach from here. I am given ...
7
votes
3answers
219 views

How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$?

How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$ in the $C^2(0,\infty)$ space solutions?
7
votes
2answers
247 views

Solutions to the equation $y^{(n)} y = 1$ for even $n$

A long time ago I was curious about the closed-form solutions to the equation: \begin{equation*} \frac{d^{n}y}{dx^n} y = 1. \end{equation*} For $n$ an odd number, try $y = A x^k$. Then $y^{(n)} = A ...
7
votes
1answer
189 views

Nonlinear 1st order ODE involving a rational function

$$y'=\frac{-6x+y-3}{2x-y-1}$$ Is there a foolproof method for tackling equations of the form $y'=\dfrac{ax+by+c}{dx+ey+f}$ ? I've tried a few substitutions (like $y=vx$ and $v=2x-y-1$, neither of ...
7
votes
2answers
9k views

Plotting Differential Equation Phase Diagrams [closed]

I haven't got Matlab, nor have I found a suitable online tool. Could someone plot the phase diagram for the following, or point me in the right direction? $$\frac{dx}{dt} = y - x, \frac{dy}{dt} = x(4 ...
7
votes
2answers
18k 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 ...
7
votes
2answers
416 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 ...