Question

if dy/dx is inversely proportional to yand y=2 when x=0 and y=4 when x=2, find y when x=3. does the mod in ln|x| go away??

Answer 1

given
dy/dx = k/y where k is constant of proportionality

Answer 2

so separating the variables we have

ydy=kdx

Answer 3

yeah and then i got x=kln|y|+c

Answer 4

no
integrating ydy=kdx
(y^2)/ 2= kx +c where c is integrating constant

Answer 5

whenx x=0 y=2 implies c=2

Answer 6

okay conitnue?

Answer 7

now when x=2 then y=4
that means
16/2 =2k+2
implies k=3

Answer 8

OH WAIT ONE SECOND. WROTE QUESTOIN WRONG. SORRY

Answer 9

its dx/dy=k/y not dy/dx

Answer 10

hence (y^2 )/2 = 3x +2
or y^2=6x+4

Answer 11

but you have posted dy/dx

Answer 12

yeah my bad .. sorry i typed wrong its dx/dy

Answer 13

then dx/dy = k/y is it correct

Answer 14

again sepearte the variables
dx = kdy/y

Answer 15

yup

Answer 16

x=ln|y|k +c ?

Answer 17

or is it just lny is there mod ?

Answer 18

x= k(ln|y|-ln|c|) instead of C we can use ln|c|

Answer 19

howcome??

Answer 20

ln|y/c|=x/k

Answer 21

for constants we can use c or logc or sinc as per depending on the question to make work easier

Answer 22

or
y=ce^(x/k) when x= 0 y=2 implies c=2

hence
y=2* e^(x/k)

Answer 23

ln|y/c|=x/k
or k = x/ ln|y/c| now when x=2 y=4 and alsoc c=2
hence
k= 2/ ln|4/2|= 2/ln2