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answer choices
y=Cx^2
x=Cy^2
x^2 – y^2 = C^2
x2 + y2 = C^2

okay so i know i have to reverse engineer this but i'm not doing too well

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Could you not let x = 1 and y = 1 and then set it up as dy/dx that way you can see the slope easier?

http://www.mathscoop.com/calculus/differential-equations/slope-field-generator.php
this might help

okay can you walk me through your answer ?

Easy to see that they are family of parabolas that open up, under vertical translation

lol i know i dont know how to make the constant multiply :P

x = Cy^2
1 = C2yy'
x/1 = y/(2y')
y' = y/(2x)

okay and if we graph that we get

since you have "one" arbitrary constant, you want to differentiate the solution just once

okay
so from x = Cy^2
to 1 = C2yy'
the only thing you did was derive each side right ?

right, just implicitly differentiate both sides with respect to x

so what did you do from 1 = C2yy'
to x/1 = y/(2y')
?

x = Cy^2 ---------------(1)
1 = C2yy' --------------(2)
next divide : (1)/(2)

okay umm why ?

your goal is to get rid of that arbitrary constant C
dividing both equations does just that

Nice explanation @ganeshie8

okay i see so what do you do after you vet rid of the c ?

get*

okay so let me try to do that for a different answer choice and tell me if i'm way off.

the differential equation y' = y/(2x) represents the whole family of curves x = Cy^2

Btw, none of the options are correct, but if I were you, I would tick option A

um okay what makes you chose option 1 ?

okay sounds about right :P , will let you know if its right in a little while when i turn it in

good luck! please do share what the grader thinks is right

will do ! thanks for all your help guys ! :D

okay are you suggesting its option 1 then ?

i'd go A
i can justify it, just lost a load of latex, so i might try post it again later

i was right !! :D thanks again !!

i reckon the "actual" right answer is:
\(y=\frac{x^2}{2}+C \)

i mean it not i