2
votes
0answers
63 views

Please identify this equation: $\nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A$

Is this equation $$ \nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A $$ somehow named? F and A are vector fields. I guess inhomogeneous sign reversed Helmholtz equation isn't appropriate ...
1
vote
3answers
49 views

Definition of homogeneous ODE

In my lecture notes, it gives this following definition of a homogeneous ODE: A differential equation is called homogeneous if it can be written in the form $x′=f(\frac{x}{t})$ Then in one of ...
0
votes
0answers
22 views

“Differential variations”?

This passage in an old book on trigonometry calls these relations among parts of a spherical triangle "differential variations". The "parts" are three sides and the three angles; when the sides are ...
1
vote
2answers
188 views

Independent and Dependent Variable Meaning?

Given a differential equation, for example: $\displaystyle \frac{d^2x}{dt^2} + a \frac{dx}{dt} + kx = 0$ Is there a reason why we call $x$ the dependent variable and $t$ the independent ...
0
votes
1answer
32 views

Given a solution $y(z)$ of a linear ODE, what is the general name for other independent solutions?

It is a well known fact that solutions of a linear ODE form a vector space whose dimensions equals the order of the differential equation. Given $y(z)$ is a solution of a given ODE of order $n$, how ...
0
votes
1answer
51 views

Correct name for equation with negative delay

How to correctly call the following equation: $$ \dot x(t) = F \big(x(t), x(t+\tau)\big), \quad \tau > 0. $$ I know, that the equation $$ \dot x(t) = F \big(x(t), x(t-\tau)\big), \quad \tau ...
3
votes
3answers
413 views

Non-ordinary differential equation?

Does such a thing exist ? Can't seem to find anything about it so i was wondering : why bother calling something "Ordinary Differential Equation" if the "Ordinary" part doesn't bring anything to it ? ...
1
vote
1answer
116 views

Half-stable vs saddle node

What is the difference between a half-stable and a saddle node in two and three dimensions?
0
votes
3answers
4k views

What is the meaning of equilibrium solution?

What are the equilibrium solutions for the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}t} = 0.2\left(y-3\right)\left(y+2\right)$ My Question: What does equilibrium solution mean in this ...
4
votes
1answer
165 views

Eigenvalues for Sturm Liouville problems and more general ODE/PDE Problems

I'm struggling to find a geometric, or at least some intuitive understanding of eigenvalues and eigenfunctions in Sturm-Liouville problems (which I've been looking at in a PDE course). For instance, ...
4
votes
2answers
100 views

Distinction between error estimator and error indicator

When solving differential equations numerically one can incur discretization error and one can construct a posteriori error estimates to approximate the true error. There is a distinction often made ...
1
vote
0answers
64 views

Definition of fragility

What does it mean for a solution to a system of differential equations to be fragile? A context for the term can be found here: This is taken from here in Mathematical Methods for Mechanics: A ...
0
votes
1answer
65 views

What should applying the Runge-Kutta-method 4 on a differential equation be called?

What should applying the Runge-Kutta-method 4 to a differential equation using a program be called? Does it qualify as "simulation"? I'm asking because I'm writing a document for school, and now I'm ...
2
votes
4answers
989 views

$y'''-y=x^{2}$ has solution — `“multiplicity”`?

The page 667 of the book (sorry not in English) claims $y'''-y=x^{2}$ to have the solution $$y(x)=C_{1}e^{x}+e^{-x/2}\left(C_{2} \cos \left( \frac{\sqrt{3}x}{2} \right)+C_{3} ...
2
votes
1answer
5k views

Complementary Solution = Homogenous solution?

I have calculated solutions to homogenous equations but is the complementary solution mentioned here the same as the homogenous solution? Let's take example $y''-3y'+2y=\cos(wx)$ and now ...
3
votes
1answer
328 views

Why are differential equations called differential equations?

Why are differential equations called differential equations?