Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.
48
votes
3answers
63k views
Teenager solves Newton dynamics problem - where is the paper?
From Ottawa Citizen (and all over, really):
An Indian-born teenager has won a research award for solving a
mathematical problem first posed by Sir Isaac Newton more than 300
years ago that has ...
22
votes
1answer
1k views
Why isn't the 3 body problem solvable?
I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of ...
21
votes
8answers
1k views
What's so special about sine? (Concerning $y'' = -y$)
In an attempt to actually grok sine, I came across the $y''= -y$ definition.
This is incredibly cool, but it leads me to a whole new series of questions. Sine seems pretty prevalent ...
18
votes
7answers
2k views
Good 1st PDE book for self study
What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic ...
17
votes
4answers
518 views
What am I doing when I separate the variables of a differential equation?
I see an equation like this:
$$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$
and solve it by "separating variables" like this:
$$y\textrm{d}y = e^x\textrm{d}x$$
$$\int y\textrm{d}y = \int ...
16
votes
7answers
1k views
“How long 'til we get there?” Road trip puzzle
Road trips can be fun, but they often appear to go slower the closer you get to your destination. I thought up this puzzle while on a recent trip. Thought it would be good food for thought. Curious ...
16
votes
2answers
707 views
please solve a 2013 th derivative question?
$ f(x) = 6x^7\sin^2(x^{1000}) e^{x^2} $
Find $ f^{(2013)}(0) $
A math forum friend suggest me to use big O symbol, however have no idea what that is, so how does that helping?
16
votes
5answers
679 views
If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
Please help me to solve this question:
Suppose $f:[a,b] \to \Bbb R$ satisfies:
$f''(x)+f(x)>0$ and $f(x)>0$ for all $x\in(a ,b)$;
$f(a)=f(b)=0$.
Prove that $b-a>\pi$.
...
15
votes
1answer
535 views
Stochastic interpretation of Einstein Equations
Einsteins theory of gravitation, general relativity, is a purely geometric theory.
In a recent question I wanted to know what the relation of Brownian Motion to the Helmholtz equation is and got a ...
14
votes
1answer
212 views
How does one parameterize the surface formed by a *real paper* Möbius strip?
Here is a picture of a Möbius strip, made out of some thick green paper:
I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
14
votes
8answers
979 views
Proof for exact differential equations shortcut?
Today in my math class, we learned about exact differential equations. During class, our teacher first taught us the accepted way to solve exact equations, but then, told us of a shortcut that one of ...
13
votes
4answers
798 views
If $f(x)=f'(x)+f''(x)$ then show that $f(x)=0$
A real-valued function $f$ which is infinitely differentiable on $[a.b]$ has the following properties:
$f(a)=f(b)=0$
$f(x)=f'(x)+f''(x)$ $\forall x \in [a,b]$
Show ...
13
votes
3answers
263 views
Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$
Find all functions satisfying $f(2x)=2f'(x)f(x)$
Under given condition, can't we find explicit solutions?
13
votes
3answers
402 views
Find $f$ where $f'(x) = f(1+x)$
Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function such that
$$f'(x) = f(1+x)$$
How can we find the general form of $f$? I thought of some differential equations, but not sure how ...
13
votes
1answer
283 views
Hilbert's 19th problem: Why do we care?
Hilbert's 19th problem asks:
Are the solutions of regular problems in the calculus of variations always necessarily analytic?
This was proven to be true (through the work of Sergei Bernstein, ...
12
votes
5answers
278 views
Solve the sequence : $u_n = 1-(\frac{u_1}{n} + \frac{u_2}{n-1} + \ldots + \frac{u_{n-1}}{2})$
For a physical model I am trying to solve this sequence:
$$\begin{align*}
u_1 &= 1 \\
u_2 &= 1-\left(\frac{u_1}{2}\right) \\
u_3 &= 1-\left(\frac{u_1}{3} + \frac{u_2}{2}\right) \\
u_4 ...
12
votes
4answers
252 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?
12
votes
1answer
485 views
Recursive solutions to linear ODE.
When finding the solutions to the simple ODE
$$ y'- mxy= x^n \text{ ; } y(0) = 0$$
I found the following:
Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define
...
11
votes
3answers
479 views
When do the Freshman's dream product and quotient rules for differentiation hold?
This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and ...
11
votes
5answers
297 views
history of the double root solution of $ay''+by'+cy=0$
Motivation: It is a well-known fact that $ay''+by'+cy=0$ has solutions which are found from substituting the ansatz $y=e^{\lambda t}$ into the DEqn. It turns out that we replace the calculus problem ...
11
votes
2answers
309 views
Sum of derivatives of a polynomial
Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$.
Show that ...
11
votes
1answer
143 views
How find this function $f(x)$
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous such that $\dfrac{f(x)}{x}=f'\left(\dfrac{x}{2}\right)$. Find $f(x)$.
(2):if $f(x)$ on $x\in[a,b] $ be continuous,find all $f(x)$?
I think this is an ...
11
votes
2answers
232 views
Algebraic interpretation of Lyapunov functions
I have recently learned the method of Lyapunov functions to rule out periodic solutions in two-dimensional nonlinear systems. My understanding is that there is some Lyapunov function for any nonlinear ...
11
votes
1answer
208 views
When does a vector field admit orthogonal fields?
My question is:
Let $\,X$ be a nonvanishing smooth vector field over an open subset $U \subset \mathbb{R}^3$. Which conditions on $X$ guarantee the existence of a smooth nonvanishing vector field ...
11
votes
3answers
296 views
Are there “differential equations” involving derivations in the sense of abstract algebra?
There is this abstract notion of a derivation, which really only cares about the property
$$D(ab)=aD(b)+D(a)b,$$
where $a,b$ are elements of some algebra. This only tangents the ideas, which lead to ...
10
votes
8answers
2k views
how do you solve $y''+2y'-3y=0$?
I want to solve this equation:
$y''+2y'-3y=0$
I did this:
$y' = z$
$y'' = z\dfrac{dz}{dy}$
$z\dfrac{dz}{dy}+2z-3y=0$
$zdz+2zdy-3ydy=0$
$zdz=(3y-2z)dy$
$z=3y-2z$
...
10
votes
2answers
987 views
Things I must know before taking differential equations course
I intend to take this course named "Differential Equations" and per the department followings contents will be taught
...
10
votes
4answers
837 views
Can this gravitational field differential equation be solved, or does it not show what I intended?
This is the equation I'm having trouble with:
$G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$
That's the non-vector form of the universal law of gravitation on the left and Newton's second law of motion ...
10
votes
1answer
195 views
Suppose $\phi$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$. Then $\phi\in W^{2,1}$
I'm trying to prove the statement in the title in as simple a way as possible. It is Theorem 3.2.9 in Helein's book "Harmonic maps, conservation laws, and moving frames", although it is not proved ...
10
votes
1answer
452 views
What is, how do you use, and why do you use differentials? What are their practical uses?
What is a differential? And how is it useful? What is its practical use?
For example, in Electromagnetic Wave Theory as it pertains to diffraction gratings, we have an equation like this one: ...
10
votes
2answers
238 views
Solving a differential equation involving $y$ and its exponential
Hi all I have a question Ive been asked to solve. But I have no idea where to begin.
The equation is $y'=\dfrac{y+e^x}{x+e^y}$.
I think this is homogeneous but I have no idea as to how to ...
9
votes
4answers
547 views
Solving $(t^2+1)(y''-2y+1)=e^t$ with the initial conditions: $y(0)=y'(0)=1$
Since it is important to me I would like to award a user who would kindly explain me what are my mistakes and what is the correct way to solve the whole problem with 500 points.
I'd really like your ...
9
votes
6answers
2k views
How are eigenvectors/eigenvalues and differential equations connected?
In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts.
I learned that ...
9
votes
3answers
160 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 ...
9
votes
3answers
206 views
Differential equations that are also functional
I was toying with equations of the type $f(x+\alpha)=f'(x)$ where $f$ is a real function. For example if $\alpha=\frac{\pi}{2}$ then the solutions include the function $f_{\lambda,\mu}(x)=\lambda ...
9
votes
3answers
4k views
Definition of a Differential Equation?
Here is one definition of a differential equation:
"An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a ...
9
votes
3answers
252 views
Question about square-wheeled cars
It's kind of an infamous problem in differential equations to find the correct road surface so that a car with square wheels (and an axle located in the center) keeps its axle level as it drives ...
9
votes
1answer
712 views
How to compute the first eigenvalue of Laplace operator in an ellipse?
Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$.
It is a fact that the eigenvalue problem for the Laplace ...
9
votes
1answer
377 views
How can I a priori justify a power series ansatz to solve an ODE?
I am currently thinking about how to a priori justify a power series ansatz to solve a nonlinear ODE of first order.
Let, for example, $y'(x)=y^2-x^2$ with $y(0)=1$.
The right hand side of the ...
9
votes
0answers
101 views
Ramanujan style nested differential Equation
So I was exploring some math the other day... and I came across the following neat identity:
Given $y$ is a function of $x$ ($y(x)$) and
$$
y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
9
votes
1answer
335 views
Solve this equation : $y'(x)+\frac{1}{x}=\frac{1}{y}$
General and particular solution for this first-order nonlinear ODE :
$$y'(x)+\frac{1}{x}=\frac{1}{y}$$
9
votes
0answers
277 views
Osgood condition
Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
9
votes
1answer
1k views
Solving ODEs: The Frobenius Method, worked examples
I find the Frobenius Method quite beautiful, and I would like to be able to apply it. In particular there are three questions in my text book that I have attempted. In each question my limited ...
8
votes
5answers
1k views
Functions that are their Own nth Derivatives for Real n
Consider (non-trivial) functions that are their own nth derivatives. For instance
$\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$
$\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$
...
8
votes
3answers
300 views
Proof for law of complex exponents using only differential equation
I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $$e^{a+b}=e^ae^b,$$ which only uses the definition that $$y=e^{zt}$$ is a ...
8
votes
2answers
338 views
Why do books say “of course” it's never that simple, differential equations?
I'm reading about differential equations. It says that a first order ODE may be written:
M(x,y)dx + N(x,y)dy = 0
Then if there is some function f(x,y) such that df/dx = M and df/dy = N you can ...
8
votes
3answers
257 views
Nicer expression for the following differential operator
I have the following sequence of differential operators:
$$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$
Is there any expression involving a sum of "normal" ...
8
votes
3answers
172 views
8
votes
3answers
307 views
The aim in a course of differential equations?
As I used to understand the primary aim of a student learning differential equations is that given a differential equation he should be able to solve it. However while recently reading a note on the ...
8
votes
3answers
398 views
$w^4+(w')^2 = g(t)$
I have a question about a first order non-linear deferential equation. I have tried many method to solve this problem but not successful yet.
Here is my question;
$$w^4 + (w')^2 = g(t)$$
$$w' = ...
