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Messages - Victor Ivrii

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16
Home Assignment 2 / Re: problem 5 (23)
« on: January 27, 2019, 07:07:51 PM »
There is NO root. You need to parametrize before integration

17
APM346--Misc / Re: analytic extension
« on: January 26, 2019, 11:31:38 AM »
If series has an infinite radius of convergence then it converges on the whole plane. If the radius of convergence is finite ...

However, even if the radius of convergence is infinite, it does not answer to many questions. F.e. from the decomposition of $e^z$ one cannot derive that $e^x$ fast rapidly as $\mathbb{R}\ni x\to +\infty$ and rapidly decays as $\mathbb{R}\ni x\to -\infty$.

18
Home Assignment 2 / Re: Assignment 2.2 Problem3 (16)
« on: January 24, 2019, 11:43:29 AM »
Quote
Given the question $u_t + 3u_x -2u_y = xyu$. My steps are:
$$\frac{dt}{1} = \frac{dx}{3} = \frac{dy}{-2} = \frac{du}{xyu} \Rightarrow x=C_1+3t, y=C_2-2t$$
$$xydt = \frac{du}{u} \Rightarrow (C_1+3t)(C_2-2t)dt = \frac{du}{u}$$
$$\ln u = C_1C_2 t + \frac{t^2}{2} (3C_2 - 2C_1) - 2t^3 +\phi(C_1, C_2)$$
Since $C_1=x-3t$, $C_2=y+2t$ we plug
$$
u= f(x-3t, y+2t) \exp \bigl( (x-3t)(y+2t)t +\frac{t^2}{2}(3y+6t-2x+6t) -2t^3\bigr)
$$
with an arbitrary function $f.,.)$ of two variables ($f=e^\phi$).

19
Home Assignment 2 / Re: Assignment 2.2 Problem3 (16)
« on: January 24, 2019, 06:16:36 AM »
What you really need to do is to replace $C_1,C_2$ by their expression through $x,y,t$

20
Home Assignment 2 / Re: Home Assignment 2 Problem 2(a)
« 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.

21
Home Assignment 2 / Re: Concept: Where does the (-) come from
« on: January 23, 2019, 07:10:24 AM »
Yes, I do mean that equation. When I try it myself however, (lines 2 to 3 of my initial post) the math comes out that dx = -tdt.
You need to read Section 2.1 of textbook to understand that equation $au_t+bu_x=f$ requires equation of integral curves $\frac{dt}{a}=\frac{dx}{b}=\frac{du}{f}$. What is on your post are incomprehensibly  written senseless manipulations.

22
Home Assignment 2 / Re: Concept: Where does the (-) come from
« on: January 22, 2019, 04:49:49 PM »
Do you mean $u_t + tu_x=0$? Then $\frac{dt}{1} = \frac{dx}{t}=\frac{du}{0}$. The first equation implies $$dx=tdt \implies x=\frac{1}{2}t^2+C\implies x-\frac{1}{2}t^2=C.$$

23
Home Assignment 2 / Re: problem 5 (23)
« on: January 20, 2019, 12:32:48 PM »
I obtain solutions for all 4 subproblems, except two of them with arbitrary function.
No, your solution is incorrect because it is not a continuous single-valued function. I gave you a hint: what is the natural parameter along integral curves?

24
Home Assignment 2 / Re: problem 5 (23)
« on: January 20, 2019, 04:47:13 AM »
Sure, $x$ and $y$ are not independent along integral curves. To proceed you need to parametrize the integral curve. Think: what is the best way to parametrize it?

25
Home Assignment 2 / Re: problem4 (20)
« on: January 19, 2019, 07:03:34 AM »
Please learn how to post math properly  Also, asking for help, copy the problem.

26
Home Assignment 2 / Re: Home Assignment 2 Problem 2(a)
« 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.



27
Home Assignment 2 / Re: Secondary Textbook Chp1.2 Exercise 10
« on: January 18, 2019, 09:05:21 PM »
Read Section 2.1, Subsection  "semilinear equations"

28
Home Assignment 1 / Re: Home Assignment 1
« on: January 16, 2019, 04:57:59 PM »
Junjing, you are right but I am not sure if anyone but me would be able to read your solution

29
Home Assignment 2 / Re: Home Assignment 2 problem 1(a)
« on: January 15, 2019, 05:04:28 PM »
You draw just a bit more than one period and say "$\pi$-periodic"

30
APM346--Misc / Re: physical interpretation
« on: January 15, 2019, 05:02:51 PM »
Physical interpretation is useful. However Quizzes, Tests and Exam neither test nor require any knowledge of Physics. Like in ODE or Calculus I, II classes.

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