Differential Eqns
@diff_eq
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Tweets on ordinary and partial differential equations from @JohnDCook
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Laplace transforms turn differential equations into algebraic equations.
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The math to go to the moon was largely differential equations.
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For over 300 years, one of the most fruitful ideas in applied mathematics has been to model systems by differential equations.
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Today is the initial condition for the rest of your life.
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Laplace transforms turn differential equations into algebraic equations.
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Maxwell's system of partial differential equations for electricity and magnetism
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'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
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Laplace transforms turn differential equations into algebraic equations.
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'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
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'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
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Why are PDEs called elliptic, parabolic, and hyperbolic?
Quick and dirty answer.
u_xx + u_yy = f is sorta like x² + y ² = r ².
u_t = u_xx is sorta like y = x ².
u_tt - u_xx = f is sorta like x² - y ² = c.
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'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
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Hamiltonian: The new Broadway hit about the sum of potential and kinetic energy.
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The big three PDEs are Laplace's equation, heat equation, and wave equation. Canonical examples of elliptic, parabolic, and hyperbolic.
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'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
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376
'Never in my life have I heard of anyone solving a first order differential equation by finding an integrating factor.' -- Gian-Carlo Rota
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Laplace transforms turn differential equations into algebraic equations.
6
64
365
The first thing people did with early electronic computers was numerically solve differential equations.
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38
356
The most common differential equations in application are 2nd order, because Newton's laws are 2nd order.
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37
350
If f(x + iy) = u(x,y) + i v(x, y) is analytic, then u and v satisfy the Cauchy-Riemann equations: u_x = v_y, v_x = -u_y.
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If a linear ODE has complex eigenvalues, solutions oscillate. If real parts are negative, oscillations decay exponentially.
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It's OK to use dubious math to produce a solution to an ODE as long as you rigorously show that the solution is correct.
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20
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'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
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If a linear ODE has complex eigenvalues, solutions oscillate. If real parts are negative, oscillations decay exponentially.
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32
313
Noether's theorem: If a system has continuous symmetries, there are corresponding quantities conserved over time.
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The big three PDEs are Laplace's equation, heat equation, and wave equation. Canonical examples of elliptic, parabolic, and hyperbolic.
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42
301
The Laplacian operator is the divergence of the gradient.
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44
294
Laplace transforms can turn delay differential equations into algebraic equations.
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Laplace transforms turn differential equations into algebraic equations.
7
49
270
Laplace transforms turn differential equations into algebraic equations.
5
38
273
'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
9
79
257
It's OK to use dubious math to produce a solution to an ODE as long as you rigorously show that the solution is correct.
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28
254
An equation of the form M(x) + N(y)y' = 0 is called separable. Write as M dx = - N dy and integrate both sides.
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You can usually apply initial conditions as an afterthought, but boundary conditions have to be baked in from the beginning.
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Differential equations date back almost as far as calculus, to the late 1600's.
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'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
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You can turn an nth order ODE into a system of n first order ODEs by introducing new variables.
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Extended Kalman filter in Python with System Identification
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If f(x + iy) = u(x,y) + i v(x, y) is analytic, then u and v satisfy the Cauchy-Riemann equations.
2
22
245
Laplace transforms can turn delay differential equations into algebraic equations.
3
32
239
'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
6
57
233
The first thing people did with early electronic computers was numerically solve differential equations.
7
21
237
Noether's theorem: If a system has continuous symmetries, there are corresponding quantities conserved over time.
3
45
221
If f(x + iy) = u(x,y) + i v(x, y) is analytic, then u and v satisfy the Cauchy-Riemann equations: u_x = v_y, v_x = -u_y.
10
24
226
It's OK to use dubious math to produce a solution to an ODE as long as you rigorously show that the solution is correct.
6
18
229
The Laplacian operator is the divergence of the gradient.
4
17
229
Differential equations date back almost as far as calculus, to the late 1600's.
3
16
214
There's a growing sentiment that methods of computing differential equations should not be taught anymore because they're nothing but tricks.
But these "tricks" foreshadow a lot of other mathematics, e.g. Lie group theory. Exploiting symmetry is not a mere trick.
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Neural Ordinary Differential Equations
Technical paper:
Pop overview:
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The Laplacian operator is the divergence of the gradient.
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26
216
"A small error in the beginning of something is a great one at the end." -- Thomas Aquinas
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64
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Differential equations date back almost as far as calculus, to the late 1600's.
0
34
210
Laplace transforms turn differential equations into algebraic equations.
5
42
214
Noether's theorem: If a system has continuous symmetries, there are corresponding quantities conserved over time.
4
28
220
To solve y' + p(x) y = g(x), multiply both sides by the integrating factor exp(P(x)) where P'(x) = p(x). Then integrate.
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212
'The partial differential equation entered theoretical physics as a handmaid, but has gradually become mistress.' -- Einstein
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'The history of transistors is the history of solving Schrödinger's equation in various materials.' -- Leon Lederman
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32
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Noether's theorem: If a system has continuous symmetries, there are corresponding quantities conserved over time.
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36
206
Bernoulli's equation: y' + p(x) y = q(x) y^n. Can be turned into a linear equation by the substitution u = y^(1-n).
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If a linear ODE has complex eigenvalues, solutions oscillate. If real parts are negative, oscillations decay exponentially.
4
34
204
Differential equations date back almost as far as calculus, to the late 1600's.
2
22
193
It's OK to use dubious math to produce a solution to an ODE as long as you rigorously show that the solution is correct.
7
26
201
Laplace transforms turn differential equations into algebraic equations.
1
47
197
Start of solving the heat equation by separation of variables
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28
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The first thing people did with early electronic computers was numerically solve differential equations.
4
20
196
Bernoulli's equation: y' + p(x) y = q(x) y^n. Can be turned into a linear equation by the substitution u = y^(1-n).
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30
190
'In order to solve a differential equation you look at it till a solution occurs to you.' -- George Polya
6
48
189
'The partial differential equation entered theoretical physics as a handmaid, but has gradually become mistress.' -- Einstein
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'Classical mathematics concentrated on linear equations for a sound pragmatic reason: it could not solve anything else.' -- Ian Stewart
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The Yang-Mills equations are the Euler-Lagrange equations of the Yang-Mills Lagrangian.
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The most common differential equations in application are 2nd order, because Newton's laws are 2nd order.
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'Knowing what is big and what is small is more important than being able to solve partial differential equations.' -- Stan Ulam
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“Why are we spending so much time on pendulums?”
“Because a lot of things act sorta like pendulums. Skyscrapers, for example.”
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'Classical mathematics concentrated on linear equations for a sound pragmatic reason: it could not solve anything else.' -- Ian Stewart
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32
178
'The history of transistors is the history of solving Schrödinger's equation in various materials.' -- Leon Lederman
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To solve y' + p(x) y = g(x), multiply both sides by the integrating factor exp(P(x)) where P'(x) = p(x). Then integrate.
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17
175
The big three PDEs are Laplace's equation, heat equation, and wave equation. Canonical examples of elliptic, parabolic, and hyperbolic.
2
42
171
To solve y' + p(x) y = g(x), multiply both sides by the integrating factor exp(P(x)) where P'(x) = p(x). Then integrate.
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24
171
'The fact is that PDEs, in particular those that are nonlinear, are too subtle to fit into a too general scheme. On the contrary, each important PDE seems to be a world in itself.' -- Sergiu Klainerman
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