Questions tagged [differential-equations]

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3
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1answer
97 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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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
78 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?
3
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1answer
234 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 ...
2
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1answer
92 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 ...
5
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1answer
140 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
117 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 ...
-1
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1answer
52 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 ...
5
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2answers
347 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 $a=b=c=0$...
6
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3answers
111 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? ...
2
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1answer
173 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 ...
0
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0answers
47 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. ...
0
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1answer
280 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 ...
4
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2answers
215 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 ...
6
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1answer
374 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 ...
6
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3answers
239 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 ...
3
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1answer
154 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 ...
6
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2answers
201 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 ...
7
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4answers
777 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 ...
8
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1answer
664 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?