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Questions tagged [differential-equations]

DO NOT USE THIS TAG just because the question contains a differential equation!

DO NOT USE THIS TAG just because the question contains a differential equation!

Almost all physical systems are governed by differential equations. Hence it is often a poor way to classify a question to use this tag.

865 questions
18
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6 answers

Any physical example of an "explosive" differential equation $ y' = ky^2$?

I was told that in physics (and in chemistry as well) there are processes that may be described by a differential equation of the form $$ y' = ky^2. $$ That is, the variation of a variable depends from the number of pairs of the elements. I…
mau
  • 291
4
votes
5 answers

Why differential equations?

Natural phenomena (e.g. heat flow) and systems (e.g. electrical circuits) are usually described using differential equations. Why is that? Also, usually people use "constant coefficients linear differential equations" of low order (one or two,…
Learn_and_Share
  • 165
3
votes
2 answers

Should every physical problem formulated as a differential equation have a mathematical solution?

I encountered the following statement in Boyce's Elementary Differential Equations and Boundary Value Problems : Not all differential equations have solutions; nor is the question of existence purely mathematical. If a meaningful physical problem…
StefanGill
  • 95
3
votes
2 answers

What does it mean to "solve an equation"?

I don't understand what is meant by there being a "solution" to an equation. For example, what does a solution to the wave or heat equation represent, and what are we solving for? Of course, we can use such equations to perform computations, but…
Couchy
  • 188
2
votes
1 answer

Solving 2nd-order ODEs

I was reading this PDF REF per request: Title: Fourier Series: The origin of all we’ll learn Link: http://www.math.binghamton.edu/paul/506-S11/CpxFn2.pdf Author: Paul Loya which I found very good to explain the origin of the Fourier series.…
user18490
  • 203
2
votes
1 answer

A general solution to continuity equation

Let us write the standard continuity equation $$\frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \vec{\jmath} = 0.$$ Should the relation $\vec{\jmath} = \rho \vec{v}$ be considered as a general solution of the continuity equation? If so, how to…
user17116
1
vote
0 answers

How to write the general solution to the Laplace or Helmholtz equation for a 3D finite cylinder?

I have a finite cylinder with the solution satisfying the Laplace equation outside. Using separation of variables in cylindrical coordinates $(r, \phi, z)$ for the 3D Laplace equation, I get (with separation constants $m,s$) \begin{align} f(r, \phi,…
user911fas
  • 23
1
vote
0 answers

Set of first order ODE

What's the condition for $\dot{x} = f(x,y)\\ \dot{y} = g(x,y)$ To be rewritable as $\dot{x} = \frac{\partial F(x,y)}{\partial y}\\ \dot{y} = -\frac{\partial F(x,y)}{\partial x}$ Can I always find a transformation $z=z(x,y)$ such that $\dot{x} =…
Bondo
  • 137
0
votes
1 answer

On different methods of solving differential equations

I've studied the basic concepts of partial differential equations, and one question comes to my mind. What are the propuse of the diferent methods of resolution of differential equations. For example if you start…
Álvaro Rodrigo
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1 answer

Can solutions of the Poisson's equation be written as linear combinations of Laplace's equation solutions?

Given that the Laplacian operator $\Delta$ acts on the space of functions(at least $C^2$), does the equation $\Delta\phi=0$, define a base of that space such that solutions of $\Delta\psi=f$ can be decomposed in that base.
electron selectron
  • 1
-1
votes
2 answers

Application for differential equation of higher order

We found some interesting insights in differential equations of the form $y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$, i.e. for ordinary differential equations of $n$-th order with $n\geq2$. The function $F$ is polynomial which can include…
DGLL
  • 1
-2
votes
1 answer

Why two terms in General solution to second order, linear, ordinary, homogeneous differential equation?

In a second order linear homogeneous differential equation of the form: $$ ay''(t)+by'(t)+cy(t)=0 $$ the general solution is: $$ c_1y_1(t)+ c_2y_2(t) $$ Here both $y_1(t)$ and $y_2(t)$ are solutions then why both are added to form new solution that…
Manikandan C
  • 21
-2
votes
1 answer

Do delay differential equations (DDEs) ever describe real-world phenomena?

I've recently become interested in DDEs, but I don't know much about them. A DDE has the form $$\begin{align*}\dot{x}(t) = f(t, x(t - \tau)) && \tau > 0\end{align*}$$ My understanding from the readings I've done is that they are used to approximate…
user541686
  • 4,131