Question

Find the solution to the differential equation, subject to the given initial condition.

Answer 1

\[\frac{ dy }{ dx }+\frac{ y }{ 9 }=0, y(0)=16\]

Answer 2

Did you try separation of variables.

Answer 3

yeah i got dy+y/9=dx
but im not really sure what to do after that

Answer 4

I don't think you did the algebra there correctly.

Answer 5

Why don't you try writing as f(x) dx=g(y) dy

Answer 6

This is what I mean by separation of variables

Answer 7

Ah. I have no idea what I'm doing then.

Answer 8

Do you know how to undo the addition to isolate the dy/dx part?

Answer 9

\[\frac{dy}{dx}+\frac{y}{9}=0\]

Answer 10

You see addition you subtract.

Answer 11

can you do that?

Answer 12

so just subtract y/9 from both sides?

Answer 13

That is how you get dy/dx by itself
now you want g(y)dy=f(x)dx

Answer 14

soo dy/y=-1/9 dx?

Answer 15

that's right

Answer 16

now integrate both sides

Answer 17

okay, i'll try it. :)

Answer 18

is ln(y)=-1/9x+c correct?

Answer 19

now use the initial condition to find C
you are given when x is 0 y is 16. Use that to find C.

Answer 20

c=ln(16)

Answer 21

yep

Answer 22

dont I need to plug that back into the equation somewhere though?

Answer 23

Yep C

Answer 24

sooo ln(y)=-1/9x+ln(16)? i put that into the homework and it said it was wrong

Answer 25

It might want you to solve for y.
Also I would write it as ln|y|=-x/9+ln(16)
I don't know if it would count off for not having those | | instead of ( ).

But yeah if wants you to solve for y then solve for y.

Answer 26

I don't even know how to go about doing that. ;_;

Answer 27

So you don't know the inverse of natural log function?

Answer 28

I don't remember it, no.

Answer 29

what about the exponential function?

Answer 30

uh wat.

Answer 31

\[e^{\ln|x|}=x\]

Answer 32

the exponential ( where base is e) function and natural log function are inverses

Answer 33

ohhh yes, okay I remeber that now.

Answer 34

use that to solve for y

Answer 35

\[\text{ if } \ln|y|=f(x) \text{ then } y=e^{f(x)}\]

Answer 36

steffw have you got that part?

Answer 37

Once you do that. I will show you how to simplify the equation you get once you show me you have gotten that far.

Answer 38

oops, i realized i was trying to solve for the wrong thing. xD
i ended up with y=16e^-1/9x
Thanks!

Answer 39

omg you did
good :)