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Messages - Nan Choi

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MAT244--Lectures & Home Assignments / Describing Direction Fields
« on: September 20, 2018, 04:26:38 AM »
To what extent are we expected to describe direction fields?

For example, consider the differential equation from the homework question:

Section 1.1 #28

$y' = e^{-t} + y$

I used the online ODE plotter to determine its behaviour.

As $t → ∞$, $y$ diverges from $0$.
$y$ approaches $∞$ when $t > 0$, and $y$ approaches $-∞$ when $t < 0$.
At $t = 0$ and $y = -1$, the value for $y'$ is zero and the slope line is horizontal.

The question asks us to determine the behaviour of $y$ as $t → ∞$, and whether this behaviour depends on the initial value of $y$ at $t = 0$. Is the explanation above enough?

MAT244--Lectures & Home Assignments / Re: Section 2.2 Q32
« on: September 20, 2018, 03:48:04 AM »
$e^c$ is just some arbitrary constant which the textbook has replaced with just $c$. Your answer and the textbook's answer are the same.

MAT244--Lectures & Home Assignments / Re: substitution in integral
« on: September 10, 2018, 10:53:34 PM »
This is a really basic way to look at it:

Since $w'(t)$ means the derivative of $w$ with respect to $t$, we can write:

$w'(t) = \frac{dw}{dt}$

So, multiplying both sides by $dt$ gives:

$w'(t)dt = dw$

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