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

learn more… | top users | synonyms (1)

8
votes
3answers
570 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
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
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
1answer
223 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
116 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
130 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
92 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
300 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
281 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
503 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
76 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
590 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
121 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
166 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
343 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
232 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
257 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
301 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
568 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
323 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
852 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
417 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
414 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
961 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
357 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
230 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
233 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
288 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
446 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
188 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
16k 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
412 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 ...
7
votes
5answers
305 views

Solving $P(x,y)dx + Q(x,y)dy =0$: interpretation in terms of forms

I asked a similar question here which I will formulate more sharply: When we write a differential equation as $P(x,y)dx + Q(x,y)dy = 0$, what is the interpretation in terms of differential forms? ...
7
votes
1answer
200 views

prove that the following function is: $f(x) = 0$

let $f: [0,1] \to \mathbb R$ , $f$ is differentiable $f(0) = 0$ $|f'(x)|\le|f(x)|$ for $x\in [0,1]$ prove that $f(x) =0$ for $x\in [0,1]$ i believe that i need to somehow use the ...
7
votes
1answer
177 views

solution of $y' = \exp \left(-\frac yx\right) + \frac yx$

Could you help me to solve equation $$y' = \exp \left(-\frac yx\right) + \frac yx;\quad y(e) = 0$$ I know how to solve 1st order linear de like $y' = \exp \bigl(-\frac 1x\bigr) + \frac yx$ but here ...
7
votes
2answers
332 views

Is the Laplacian surjective on $C_0^{\infty}$?

Let $M := C_0^{\infty}(\mathbb{R}^n)$ denote the smooth maps with compact support. Then we have a map $\Delta:M\rightarrow M,\,\, f\mapsto \Delta f$, where $\Delta f = \sum_{i=1}^{n} ...