Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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128 votes
9 answers
23k views

Prove that $C e^x$ is the only set of functions for which $f(x) = f'(x)$

I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $ e^x$ is the same as the function itself....
116 votes
1 answer
3k views

Solving Special Function Equations Using Lie Symmetries

The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's ...
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101 votes
6 answers
5k views

Does a non-trivial solution exist for $f'(x)=f(f(x))$?

Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind ...
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99 votes
0 answers
2k views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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90 votes
1 answer
3k views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc... & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
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76 votes
4 answers
80k 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 ...
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70 votes
2 answers
15k views

Is it mathematically valid to separate variables in a differential equation? [duplicate]

I read the following statement in a book on Calculus, as part of my mathematics course: Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting integration ...
68 votes
5 answers
8k 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 e^x\textrm{d}x$...
64 votes
6 answers
8k views

Is there a reason it is so rare we can solve differential equations?

Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously ...
63 votes
15 answers
15k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
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63 votes
2 answers
68k views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...
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51 votes
2 answers
6k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
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50 votes
3 answers
3k views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
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50 votes
7 answers
4k views

What is the optimal path between $2$ fixed points around an invisible obstructing wall?

Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). ...
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49 votes
4 answers
6k views

How do we know that we found all solutions of a differential equation?

I hope that's not an extremely stupid question, but it' been in my mind since I was taught how to solve differential equations in secondary school, and I've never been able to find an answer. For ...
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47 votes
5 answers
29k views

Links between difference and differential equations?

Does there exist any correspondence between difference equations and differential equations? In particular, can one cast some classes of ODEs into difference equations or vice versa?
45 votes
7 answers
9k views

Function whose third derivative is itself.

I'm looking for a function $f$, whose third derivative is $f$ itself, while the first derivative isn't. Is there any such function? Which one(s)? If not, how can we prove that there is none? Notes: ...
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44 votes
5 answers
5k views

When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?

The question is: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution? Edit I should add ...
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43 votes
4 answers
13k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
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42 votes
5 answers
244k views

Linear vs nonlinear differential equation

How to distinguish linear differential equations from nonlinear ones? I know, that e.g.: $$ y''-2y = \ln(x) $$ is linear, but $$ 3+ yy'= x - y $$ is nonlinear. Why?
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41 votes
2 answers
3k views

Do you lose solutions when differentiating to solve an integral equation?

The question is more general but here's the problem that motivated it: I want to find all solutions to the integral equation $$f(x) + \int_0^x (x-y)f(y)dy = x^3.$$ Differentiating twice with respect ...
40 votes
3 answers
2k views

Why don't these ODEs produce the same result?

I am relatively new to differential equations, and the following problem is confusing me. Consider, for example, the ODE $x'+x=0$ such that $x(0)=1$. This has solution $x(t)=e^{-t}$. But consider an $\...
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40 votes
3 answers
6k views

What is the rigorous definition of $dy$ and $dx$?

Some background: I am a third year undergrad. I have completed two courses on Real Analysis (I have studied $\epsilon$-$\delta$ definitions of limit, continuity, differentiability, Reimann integration ...
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39 votes
2 answers
8k 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 ...
  • 5,889
37 votes
4 answers
128k 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 ...
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36 votes
7 answers
17k views

Separable differential equations: detaching dy/dx [duplicate]

I'm just learning about differential equation separability. I understand what a derivative is. One notation for derivative is $\frac{dy}{dx}$, which - misleadingly - is not a fraction. Since it's not ...
36 votes
3 answers
5k views

How is Category Theory used to study differential equations?

I know that one can use Category Theory to formulate polynomial equations by modeling solutions as limits. For example, the sphere is the equalizer of the functions \begin{equation} s,t:\mathbb{R}^3\...
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36 votes
3 answers
22k views

What is the idea behind Green's function? What does it do?

I have an exam on ordinary and partial differential equations in a couple of days and there is one concept that I am really struggling with: Green's function. I have basically read every PDF-file on ...
  • 4,077
35 votes
2 answers
2k views

What are all the generalizations needed to pass from finite dimensional linear algebra with matrices to fourier series and pdes?

I've studied Linear Algebra on finite dimensions and now I'm studying fourier series, sturm-liouville problems, pdes etc. However none of our lecturers made any connection between linear algebra an ...
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32 votes
1 answer
38k views

Explanation and proof of the 4th order Runge-Kutta method

The 4th order Runge-Kutta (RK4) method is a numerical technique used to solve ordinary differential equations (ODEs) of the following form $$\frac{dy}{dx} = f(x,y), \qquad y(0)=y_0$$ It gives $y_{i+1}$...
  • 1,460
32 votes
1 answer
3k views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
31 votes
2 answers
15k views

Can anyone explain the intuitive meaning of 'integrating on both sides of the equation' when solving differential equations?

For solving differential equations, especially the ones of the form $$g(x)dx = h(y)dy$$ we solve the equation by integrating on both sides to reveal the solution. Understanding this for ...
31 votes
2 answers
6k views

Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
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30 votes
5 answers
1k views

Why does adding a term $5f'(t)$ to $5f''(t)+10f(t)=0$ cause damping?

So we have a differential equation to model an oscillator: $$5f''(t)+10f(t)=0$$ Where the initial conditions are $f(0)=0$ and $f'(0)=4$. It is given that $f(t) = \frac{2\sqrt 2}{5}\sin\sqrt2 t$. ...
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30 votes
2 answers
3k 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 it ...
30 votes
1 answer
972 views

How to solve $\dot{x} = \frac{f(x)}{\|f(x)\|}$?

How to solve the following ODE? $$\dot{x} = \frac{f(x)}{\|f(x)\|},$$ where $x : \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f : \mathbb{R}^n \to \mathbb{R}^n$ ...
  • 1,194
29 votes
9 answers
5k 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 ...
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29 votes
2 answers
5k views

Intuition of Gronwall lemma

The Gronwall lemma is a well known and very useful statement which is used in many situations, in particular in the theory of differential equations. I have seen it so many times and even the proof is ...
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29 votes
2 answers
1k views

Justifying the "Physicist's method" for ODEs using differential forms

I need some help in untangling and solving the following exercise: Let the curve $c:[a,b] \to \mathbb{R}^2, t \mapsto (t, y(t))$ be a solution for the ODE $$ y'(x) = f(x, y(x)). $$ Justify the ...
29 votes
2 answers
2k views

Solutions to $f'=f$ over the rationals

The problem is as follows: Let $f: \mathbb{Q} \to \mathbb{Q}$ and consider the differential equation $f' = f$, with the standard definition of differentiation. Do there exist any nontrivial solutions?...
28 votes
2 answers
2k 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?
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28 votes
0 answers
760 views

Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
26 votes
6 answers
3k 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 $...
25 votes
1 answer
35k 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 $$y''...
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25 votes
2 answers
3k 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 $p(x)+p&#...
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25 votes
1 answer
23k views

How do you write a differential operator as a matrix?

How do you write a differential operator as a matrix? I'm very confused. Could someone please use examples to help me understand? Preferably with first and second-order linear differentiation.
  • 275
25 votes
2 answers
597 views

Are these equations "properly" defined differential equations? (finite duration solutions to diff. eqs.)

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it. I am trying to find out if there exists any exact/accurate/non-approximated ...
  • 1,053
24 votes
7 answers
44k views

What is the essential difference between ordinary differential equations and partial differential equations?

Please forgive my stupidity. So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between them....
  • 1,123
24 votes
2 answers
971 views

Periodic orbits of "even" perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ y'&...
  • 272k
24 votes
1 answer
3k views

Intuition behind variation of parameters method for solving differential equations

I have used the variation of parameters method (and have been taught it, although not hugely in depth) and I was wondering if I've understood the intuition behind it. In particular I've been thinking ...

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