Find the domain for the particular solution to the differential equation dy dx equals the quotient of negative 1 times x and y, with initial condition y(2) = 2.

- anonymous

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- anonymous

https://gyazo.com/6439d6ba1afb5baf216c9b05cf9296f0

- anonymous

@timo86m would you know?

- anonymous

sorry haven't taken DE yet

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## More answers

- anonymous

It's cool. I appreciate you taking your time to read the question.

- anonymous

@freckles Would you happen to know?

- freckles

first find when when dy/dx does not exist
note dy/dx is -x/y
and -x/y is fine unless ....?

- freckles

think fractions are not cool if the bottom is ...

- anonymous

0

- freckles

right we don't want y to be 0...
now we need to keep this in mind after we solve the differential
equation we will try to use this
so what do you get when you solve dy/dx=-x/y

- anonymous

I got \[y = \sqrt{\frac{ -x^2 }{ 4 }+5}\]

- freckles

not sure how you got the 5...
\[y dy=-x dx \\ \frac{y^2}{2}=\frac{-x^2}{2}+C \\ \text{ I would \not attempt \to solve for } y \text{ yet } \\ \text{ but I'm going \to use the condition given \to find } C \\ \frac{2^2}{2}=\frac{-2^2}{2}+C \\ \frac{2(2)^2}{2}=C \\ C=2^2=4 \\ \frac{y^2}{2}=\frac{-x^2}{2}+4 \\ y^2=-x^2+8 \\ \text{ but } y \neq 0 \\ \text{ so } -x^2+8 \neq 0 \text{ when ? }\]

- anonymous

What are you asking exactly?

- freckles

for what x is -x^2+8=0?

- anonymous

\[\sqrt{8}\]

- freckles

or?

- anonymous

-\[\sqrt{8}\]

- freckles

|dw:1449725022181:dw|

- anonymous

-2sqrt(2) to 2sqrt(2)

- freckles

right

- anonymous

Thanks

- anonymous

Just to confirm

- anonymous

You found C then found x-intercept?

- freckles

no remember dy/dx=-x/y , we couldn't have y is 0

- freckles

by y is 0 when -x^2+8 is 0

- anonymous

Ohhh ok got it

- freckles

since if y=0 is then y^2=0
and y^2=-x^2+8

- anonymous

Can you check over another?

- freckles

maybe

- anonymous

They quick ones

- anonymous

https://gyazo.com/ba019252b62625bf2b352b7aa4b86315

- freckles

well to find critical numbers what do you usually do?

- anonymous

set equal to 0

- freckles

what do you set equal to 0?

- anonymous

When looking at a graph they correspond to min/max values

- anonymous

f'

- freckles

ok
f'(x)=0 aka the x-intercepts of f'

- freckles

however

- freckles

let's take f(x)=x^3 as an example
f'(x)=3x^2
and 3x^2=0 when x=0
but is (0,f(0)) a max or min?

- anonymous

Min

- freckles

whaT?!!!

- freckles

|dw:1449725709459:dw|

- anonymous

Would it not be a parabola? Which unless reflected has a min

- freckles

|dw:1449725732938:dw|

- freckles

no f(x)=x^3 is not a parabola

- anonymous

Were we not talking about f'?

- freckles

I wanted to find the max/min of f(x)=x^3

- freckles

if they existed

- freckles

f(x)=x^3
f'(x)=3x^2
3x^2=0 when x=0
but (0,f(0)) is neither a max or min

- freckles

this example was suppose to show that it is not necessarily true that if f'(a)=0 then you have a min or max at x=a

- anonymous

And that is exactly why I chose D

- freckles

you chose well

- anonymous

;)

- freckles

but I'm confused why you would want to find the max or min of f'(x)

- freckles

and not f(x)

- anonymous

I was confused by the way it was written.

- freckles

hmmm... okay...

- freckles

I don't know how else to write it
sorry

- freckles

the way I usually find the critical numbers of f(x)=x^3 or some other polynomial function
is to find f'(x) and solve f'(x)=0 first

- anonymous

Isn't that what we all should do?

- freckles

well the example was a polynomial...
you should always consider f'(x)=0 even if it isn't a polynomial
there are other functions where you have to consider when f'(x) does not exist

- anonymous

Got it.

- anonymous

I'd really appreciate it if you could help me with one last one that I am completely clue less on.

- freckles

question 1: consider functions f(x)=1000 and g(x)=1
is one growing faster than the other? (look at graphs of the functions if you have to)
question 2: consider function f(x)=0 and g(x)=1
is one growing faster than the other? (look at graphs)
question 3: consider function f(x)=x^2 and g(x)=x?
is one growing faster than the other? (look at graphs)

- anonymous

Q1: No
Q2: No
Q3: Yes
But where is this coming from?

- freckles

\[\lim_{x \rightarrow \infty} \frac{1000}{1}=1000 \\ \lim_{x \rightarrow \infty} \frac{0}{1}=0 \\ \lim_{x \rightarrow \infty}\frac{x^2}{x}=\infty\]
they examples for each of the cases there

- anonymous

Oh, that wasn't the question.

- freckles

what?

- anonymous

The last question I have, I created a new post for it

- freckles

ok I don't see the question
I only see question 7
but anyways good luck on the last question whatever it is
I have to go
peace

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