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Messages - Victor Ivrii

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61
Chapter 2 / Re: Help for section 2.4, questions 14-15
« on: September 29, 2020, 04:54:13 PM »
Almost done, but you need also define interval, on which solution exists; it must include $0$

62
Chapter 1 / Re: Definition for limit is infinity?
« on: September 29, 2020, 04:35:15 AM »
That's right but it is a custom to denote by $\varepsilon,\delta$ something small. In this case capital letters, say $R,M$ would be better, for arbitrarily large

63
It was explained in the lectures that solutions in the parametric form are also admissible. You could ask it on the tutorial.

64
Chapter 2 / Re: W3L3 Exact solutions to inexact equations
« on: September 29, 2020, 04:31:19 AM »
In week 3 lecture 3, we get the example $(-y\sin(x)+y^3\cos(x))dx+(3\cos(x)+5y^2\sin(x))dy=0$. We determine that this equation is not exact, but that we can make it exact by multiplying the equation by $y^2$. We then find the general solution to the new equation is $y^3\cos(x)+y^5\sin(x)=C$.

My question is why is this good enough? We didn't answer the original question. We answered a modified version which we chose specifically because it seems easier to us. Shouldn't we still find a solution to the original equation $(-y\sin(x)+y^3\cos(x))dx+(3\cos(x)+5y^2\sin(x))dy=0$? Is there some way we can "divide out" $y^2$ from $y^3\cos(x)+y^5\sin(x)=C$ to get it?
We have not modified equation, but simply multiplied it by an integrating factor. These two equations are equivalent, except as $y=0$, that means $C=0$. But $y=0$ os also a solution to the original equation. Checking this would give you a 100% correct solution, otherwise it is almost perfect

65
Chapter 1 / Re: how to solve question 19 in section 1.2 in the textbook
« on: September 27, 2020, 11:38:44 AM »

Darren, you may want to fix: replace $c_2$ by $c^2$.


Runbo, what  about division?

66
Chapter 1 / Re: how to solve question 19 in section 1.2 in the textbook
« on: September 27, 2020, 11:37:07 AM »
Darren, you may want to fix: replace $c_2$ by $c^2$.

67
Chapter 2 / Re: Question on Week3 Lec3 Integrating Factor
« on: September 26, 2020, 09:32:40 AM »
I could be wrong, but I think we just try each one until we get one that works :)
Indeed.

68
Chapter 2 / Re: Section 2.2 Question 3
« on: September 24, 2020, 01:37:33 AM »
Because in addition to solutipons you found, there is another one, namelu $y=0$

69
Chapter 2 / Re: Textbook 2.4, Example 2
« on: September 24, 2020, 01:36:34 AM »
You should have it in Calculus II

70
In this case usual cube of the sum would be the most efficient solution

71
Chapter 1 / Re: Section 1.3 Q15
« on: September 24, 2020, 01:33:32 AM »
Simple observation is enough

72
Chapter 1 / Re: Problems to 1.2 Q20
« on: September 23, 2020, 07:58:36 AM »
You ca refer to the fact that straight lines in $\mathbb{C}$ are also straight lines in $\mathbb{R}^2$ and conversely

73
Chapter 2 / Re: Mock Quiz Answer
« on: September 21, 2020, 12:28:01 PM »
It was not my intention to to check any math, so there could be that it is impossible recovering $y$ as a function of $x$. My purpose was to let you practice techical aspects

74
Chapter 2 / Re: Lec 0101 - 9/15 Question
« on: September 20, 2020, 07:49:47 PM »
One should remember that plugging $y=uy_1$ into inhimogeneous equation leaves $u'y_1$ in the left-hand expression. If you do not remember this, therefore you just do not understand the method of variations and you should reread previous slides

75
Chapter 1 / Re: Solving roots of complex numbers
« on: September 20, 2020, 07:46:18 PM »
The worst thing you can do is to use calculator to evaluate the value of, say, $\sin (4\pi/9)$ and $\cos (4\pi/9)$ numerically. But it may be useful to mention that
$\cos (4\pi/9)+i\sin (4\pi/9)$ belongs to the first quadrant and pretty close to $i$. Just draw a little picture.

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