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 ...
3
votes
1answer
70 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
0
votes
0answers
24 views

Proper equality sign for boundary value definition

Say I have a function that needs to define a boundary condition, like $f(0) = A$. In this case it is usually fairly self evident from context, that this is a requirement that $f(0)$ needs to satisfy. ...
0
votes
0answers
24 views

What does it mean that solution of DE has ''n zeros''?

It appears in the definition of disconjugate DE: A linear DE with constant coefficents on a $t$-interval $I$ is said to be disconjugate on $I$ if no solution ($\neq 0$) has $n$ zeros on $I$. Thank ...
0
votes
3answers
137 views

Solution of a differential equation having a singularity (not everywhere defined) [closed]

Remind me about ordinary differential equations, whose solutions are not everywhere defined (have a singularity). I want to remember the exact definition of a solution with singularity, which I ...
0
votes
3answers
56 views

Definition of Linear Differential Equation

I am a 13 year old self teaching myself Differential Equations from a website and a book, I came across the definition of a Linear Differential Equation but I didn't understand the definition, I ...
1
vote
1answer
79 views

Ordinary differential equations with double resonance

I want to know what is the definition of "resonance, double resonance" in ordinary differential equations with double resonance for exemple this : what it means the probleme is resonant in infity ? ...
1
vote
1answer
54 views

Definition of purely oscillatory

This is question about a term whose definition I can find anywhere. I am given to solve a differential equation and one of the questions asks to show that the solution (we are given initial data) is ...
1
vote
1answer
158 views

Meaning of no explicit time dependence

What does "no explicit time dependence" mean in this context? : A symmetry of the KdV is given by $$\tilde x=x, \tilde t=t+\epsilon, \tilde u =u$$ as there is no explicit time dependence in the ...
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 ...
3
votes
2answers
1k views

A simple explanation of differential calculus and its link to geometry?

The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
1
vote
2answers
164 views

On the definition of jets

I have some problems with the definition of jets and it would be great if someone could help me here: In many books it is written, that the $r-th$ order jet $j^r_xf$ of a smooth function $f:M ...
1
vote
1answer
517 views

Time-invariant IVP

How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. ...