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.

A differential equation is any expression that relates the values of a collection of functions and their derivatives, as well as some coefficients. One may be asked to solve (symbolically or numerically) or estimate the solution to a differential equation. Usually, they appear as the result of a mathematical model for a physical phenomenon.

The $n^{th}$ order ordinary differential equation is of the form $$F[x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, . . . ,\frac{d^n y}{dx^n}]=0$$ where $F$ is real function of its $n+2$ arguments $x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, . . . ,\frac{d^n y}{dx^n}$.

Some times we use the prime notation for derivatives as $$F(x, y, y', y'', . . . , y^{(n)})=0$$ where the notations $y'\equiv\frac{dy}{dx}, y''\equiv\frac{d^2y}{dx^2}$ and so on.


"Differential Equations" by Shepley L. Ross

"Differential Equations With Applications and Historical Notes" by G.F. Simmons


This tag is intended for ordinary differential equations, i.e., the differential equations which contain only derivatives w.r.t. one variable and not partial derivatives.

Use the tag for partial differential equation questions.

40538 questions
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Prove that $C\exp(x)$ 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 $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because…
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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 equation. Kaufman's article describes algebraic methods…
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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 Implicit Function Theorem (finite-dimensional vector…
6 answers

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 of equation? If this equation doesn't have any…
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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 here. The gist of the theory is as follows: The…
4 answers

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 baffled mathematicians ever since. The solution…
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Is it mathematically valid to separate variables in a differential equation?

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 leads to correct answer. The book also contains…
Devarsh Ruparelia
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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: $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$$ $$y^2/2 = e^x + c$$ What am I doing when I solve…
Mark Eichenlaub
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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 complicated solutions or be unsolvable. Is there some…
15 answers

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 to learn to do complex math operations on paper…
2 answers

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 easy to read. Thanks
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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 results. I am more interested in the reason or…
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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?
7 answers

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). The wall extends $1$ mile in each direction…
4 answers

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 example, take the simple differential equation…
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