Author Topic: Home Assignment 2 Problem 2(a)  (Read 871 times)

MikeMorris

  • Jr. Member
  • **
  • Posts: 6
  • Karma: 0
    • View Profile
Home Assignment 2 Problem 2(a)
« on: January 16, 2019, 06:10:30 PM »
For this question (eqn 14) I found the characteristic curve to be $\frac{x}{y} = C$. It asks what the difference is between the cases $(x,y)\neq (0,0)$ and that the solution be continuous at (0,0). From what I understand, in the former case we can simply say the solution is any funtion of one variable, $u(x,y) = \phi (\frac{x}{y})$, on the domain $(x,y)\neq (0,0)$. In the second case, we need to pick some value for $u(0,0)$ to make it continuous, and elsewhere it's the same as the other case. Is this a sufficient explanation? It feels lacking to me. Could someone offer assistance?

Zhiman Tang

  • Jr. Member
  • **
  • Posts: 6
  • Karma: 0
    • View Profile
Re: Home Assignment 2 Problem 2(a)
« Reply #1 on: January 18, 2019, 09:17:57 PM »
I think you need to find the value of u(0,0), below is my answer, but idk whether it is correct or not.
consider u(x,y) = f(x/y)
in this case, x=0 is not a problem, but y=0 is a problem
we can write u(0,y) = f(0),
to make u a contiunuos solution, we need u(0,0)=f(0) since the value of u can't jump at y=0
so u(0,0)=f(0) is the condition to make it continuos.

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2511
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Home Assignment 2 Problem 2(a)
« Reply #2 on: January 18, 2019, 09:31:56 PM »
Since integral curves are rays (straight half-lines) from $(0,0)$  the solution in the plane wit the punched out origin is any function, constant along these rays, in particular $u=f(y/x)$.

But if we want solution in the whole plane, $u$ must be continuous at $(0,0)$ and since all rays intersect there $u$ is just a constant.



Wanying Zhang

  • Full Member
  • ***
  • Posts: 21
  • Karma: 6
    • View Profile
Re: Home Assignment 2 Problem 2(a)
« Reply #3 on: January 23, 2019, 05:13:27 PM »
I wonder why the integral curves are half-lines from (0, 0)? I agree that these integral curves are straight since the equation $C = \frac{x}{y}$, but it seems no given range for either $x$ and $y$. How is the characteristic of being half determined? Any reply would be appreciated.

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2511
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Home Assignment 2 Problem 2(a)
« Reply #4 on: January 23, 2019, 08:31:16 PM »
Please all these lines tend to $(0,0)$ in one as $s\to -\infty$ ($\frac{dx}{ds}=x$, $\frac{dy}{ds}=y$) but never reach it. Please read again ODE, Chapter 7.