Question

@phi

Answer 1

write it using dy/dx
multiply both sides by dx to "move" dx up top (and on the other side)
what else should you do ?

Answer 2

integrate!

Answer 3

how would i write it with dy/dx cause it is y'

Answer 4

y' is another way to write dy/dx

Answer 5

y' means "take the derivative of y with respect to some other obvious variable ... x in this case"

Answer 6

\[ \dot{y} \] y with a dot over it means dy/dt (derivative of y with respect to time t)

Answer 7

ohh so it would be dy/dx= 4xsqrt1-y^2

Answer 8

yes. now "separate the variables"
multiply both sides by dx
and what else should you do ?

Answer 9

wait how would it loked like with the sqrt? i know how to do it over fractions

Answer 10

after that, integrate?

Answer 11

after multiplying by dx on both sides what do you get ?

Answer 12

what would it look like separated?

Answer 13

so far you have
dy/dx= 4xsqrt(1-y^2)

after multiplying by dx on both sides what do you get ?

Answer 14

dy=4xsqrt(1-y^2)dx

Answer 15

now what should you multiply to move the sqrt part ?

Answer 16

multiply each side by sqrt?

Answer 17

if you have x=y
and multiply both sides by y you get
xy = y^2

if you multiply by 1/y you get
x/y = 1
notice multiplying by the "inverse" 1/y moved the y to the other side

Answer 18

in other words you divide both sides by sqr(stuff)
on the right side you get 1
on the other side you get a fraction

Answer 19

oh so you multiply each side by sqrt1-y^2?

Answer 20

wait i think im confused hahah

Answer 21

think like this:
if I have
x* stuff
and I want to "get rid" of stuff
I divide by stuff
x * stuff/stuff
anything divided by itself is 1, so we get
x*1 or just x

and because it's an equation we also divide the other side by stuff to keep things equal

Answer 22

oh right! so divide each side by sqrt 1-y^2

Answer 23

yes

Answer 24

what is the next step ?

Answer 25

integrate?

Answer 26

yes. the right side is the power rule.

Answer 27

the left would be arcsin(y) and the right would be 2x^2

Answer 28

yes, and toss in a C

Answer 29

now solve for y (take the sin of both sides)

Answer 30

remember sin(arcsin(y)) = y
the sin and arcsin are inverses that undo the other.

Answer 31

you get
\[ y = \sin(2x^2+C) \]
notice that no matter what number you put into sin you get out a number between -1 and 1

Answer 32

okay got it, whats next?

Answer 33

part b. which is answered up above.
replace x with 0 in your solution and you get
y= sin(C)
but there is no C that will cause the sin to return 4

Answer 34

oh that was easy , your explanation helped a lot(:

Answer 35

yes. or you can leave it using arcsin
but for part b, it's easier (for me) to use the sin version

Answer 36

okay, got it , thank you!!