Questions tagged [differential-equations]

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8
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1answer
797 views

Who invented the exponential ansatz for linear differential equations with constant coefficients?

Who invented using $e^{\lambda t}$ as ansatz for solving linear differential equations with constant coefficients?
8
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3answers
940 views

Who wrote down the first differential equation?

I am just curious who was the first person to write down a differential equation? And what was this differential equation?
8
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3answers
1k views

What are the early applications of differential equations to social sciences?

I am reading Karatzas and Shreeve (Brownian Motion and Stochastic Calculus) and on page 128, at the beginning of chapter 3 on Stochastic Integrals, one reads: I can't think of any social problem that ...
8
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4answers
913 views

Who did first use the Method of Characteristics?

Which mathematician did introduce the Method of Characteristics for solving linear Partial differential equations? I some papers I saw that Saint Venant, Lagrange, Cauchy or Riemann were attributed ...
7
votes
3answers
118 views

What were the early uses of differential equations for modeling chemical reactions?

What are some of the original examples of uses of differential equations for modeling and analyzing chemical reactions, particularly those relevant to biochemistry, involving proteins and enzymes? ...
6
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3answers
254 views

source of “logistic growth”?

I've been trying to find the source of the name of the DE modelling population growth known as logistic growth, for some time: why "Logistic" ? So far all my attempts to research it have hit dead ends ...
6
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2answers
319 views

Did Newton find the trajectory of a body moving in uniform gravity under the quadratic resistance law (the ballistic problem)?

I'm very confused by contradicting accounts of a supposed solution by Newton to the problem of finding the trajectory of a projectile moving under uniform gravity against resistance that is ...
6
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1answer
383 views

Who came up with the link between the spectrum of an operator and the poles of a meromorphic function?

From Dieudonné's "History of Functional Analysis" I learned that Picard in 1893 gave a characterization of an eigenvalue of the Laplacian as the simple pole of a meromorphic function. Is there an ...
5
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2answers
407 views

History of hypergeometric equation

It is known that Gauss studied hypergeometric equation $$x(1-x) \dfrac {d^2y}{dx^2}+(c-(a+b+1)x)\dfrac {dy}{dx}-aby=0$$ I would like to know something about history of this equation: 1) If $a=b=c=0$...
5
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1answer
253 views

Earliest Instances of a Slope/Direction Field for a First-Order ODE

Background When first encountering slope fields in calculus or elementary differential equations, students often ask "What is the purpose?" A concise answer is that slope fields provide a way to ...
4
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1answer
229 views

What is the etymology of “phase space” of a dynamical system?

The state space of a dynamical system is often called a "phase space". What is the etymology of this? (Note that I'm not asking about the history of the concept, but rather about the history of the ...
4
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1answer
137 views

How did Peano prove his existence theorem without Ascoli's theorem?

In modern proofs of the Peano Existence Theorem for ordinary differential equations, Ascoli's theorem is used. Ascoli's theorem came after Peano's proof. Did Peano prove a form of Ascoli's theorem in ...
4
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2answers
230 views

Origin of the terminology “trace operator” related to boundary-value problems for PDEs

Important results in the theory of PDEs regarding boundary-value problems are trace and extension theorems. Since the trace operator (not to be confused with the trace from linear algebra) essentially ...
4
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0answers
40 views

What kinds of differential equations were solved on the early differential analyzer?

Harold Locke Hazen and Vannevar Bush created the Differential Analyzer at MIT between 1928 and 1931 to solve differential equations, according to the Wikipedia Article on the topic. I've read this ...
3
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1answer
258 views

Source of claim that Leibniz discovered separation of variables for ODEs in 1691?

Claims I'm evaluating I've read in multiple sources that Leibniz formulated separation of variables for ODEs in 1691. A couple example sources are below. Mathematical Thought from Ancient to Modern ...
3
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2answers
677 views

History of PDE's in the 19th Century

I've been asked to write an essay on whether the work on PDE's in the 19th century belonged to applied or pure mathematics. Does anyone know of any useful sources I could use?
3
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1answer
2k views

What is the origin of the term “ordinary differential equations”?

Who has first used the term "ordinary differential equation"? Is it known, why the term "ordinary" is used here? What makes an ODE "ordinary"?
3
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1answer
56 views

Who is Wanner from Rosenrock-Wanner (ROW) methods?

I've spent some time with a search engine trying to find out about Wanner, a person whose surname is mentioned in the name of Rosenbrock-Wanner (ROW) methods primarily used for iteratively solving ...
3
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1answer
102 views

From where the so-named “elastica problem” is coming from?

In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here). The problem is presented as a system of ODEs, $$ x' = \cos (\phi) $$ $$ y' = \sin (\phi) $$...
3
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1answer
173 views

Poincare's last geometric theorem

Which problem in celestial mechanics led Poincare to his conjecture about fixed points of area preserving maps of the annulus to itself? I believe he was working on some differential equations. What ...
3
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0answers
53 views

Asymptotically Periodic Potentials

Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials? I heard it was Pierre Louis Lions, ...
2
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1answer
99 views

Does the “O” in the google doodle for Olga Ladyzhenskaya have anything to do with her work?

Ladyzhenskaya is famous for fluid dynamics and partial differential equations, both of which are beyond my pay grade. And she worked on the Navier-Stokes equations. Does this circle with the arrows ...
2
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1answer
208 views

D'Alembertian symbol $\Box$

The D'Alembertian is a generalization of the Laplacian operator to a space of arbitrary dimension and metric. Where does the D'Alembertian symbol $\Box$ come from? According to Wikipedia it has to ...
2
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1answer
103 views

Why was Indicial equations named so?

In ODE, in Frobenius method, there's an equation called "Indicial Equation." Is there any particular contextual/historical reason that it is named so?
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0answers
254 views

History of PDE's in the 19th Century (part 2)

This is a follow up to this question: History of PDE's in the 19th Century The question I have been given to answer is: The history of partial differential equations in the 19th Century belongs ...
0
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1answer
356 views

How did Newton write his equations?

Once, after a lecture, my professor of differential equations said, that Newton did not use derivatives in his work as we do today. He told us that Newton rather used some series expansions for his ...
0
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1answer
59 views

Invariance principle for stability in the sense of Lyapunov

On Wikipedia this article about the invariance principle and article states that The general result was independently discovered by J.P. LaSalle (then at RIAS) and N.N. Krasovskii, who published ...
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64 views

The Original Proofs of The Stable Manifold Theorem

The book "Differential Equations and Dynamical Systems" by Lawrence Perko says that the first proofs of the Stable Manifold Theorem are from Hadamard in 1901, Perron in 1928, and Liapunov and Perron. ...
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1answer
56 views

Why are partial derivatives necessary when deriving the equation for a vibrating string?

AFAIK, partial derivatives made it to the forefront as a result of coming up with an equation for a vibrating string i.e., 1D wave equation. From purely a physical phenomena to math mapping POV: Why ...