0
votes
1answer
20 views

Blowup of ODEs in the presence of local Lipschitzianity?

Pardon me if the question is trivial, but I am failing to decide it. Assume that we are given an ODE system $\dot{x} = f(x)$ with positive initial conditions $x(0)$ and know that $f$ is locally ...
2
votes
1answer
60 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
0
votes
2answers
262 views

Vector field with bounded integral curves

I am thinking about smooth vector fields on some (open set of an) euclidean space $\mathbb{R}^n$. I know that the integral curves of a general vector field $X$ are not defined for every time $t\in ...
5
votes
3answers
2k views

Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
2
votes
1answer
460 views

Simple proof that non-linear DE's do no not satisfy superposition?

I'm wondering if there's a simple proof that solutions to non-linear differential equations do not satisfy the superposition principle? Some explicit examples would also be great. Cheers!
3
votes
4answers
331 views

Examples of parameter dependent ODEs

I would like to have some examples of simple parameter dependent ODEs. I would like the solutions to have some physical meaning. I'll give one example so it clear what I am after: Example 1: The ...
0
votes
2answers
243 views

One dimensional boundary value problems showing interesting behaviour

I want to show some users of a piece of software some solutions of one-dimension boundary value problems (can also be initial value problems). I'm after a collection of problems whose solutions are ...
9
votes
6answers
4k views

How are eigenvectors/eigenvalues and differential equations connected?

In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...