Question

write a differential equation in dy/dt=ay+b form with all solutions approaching y=3 as t-> infinity

Answer 1

no im dumb

Answer 2

Actually, we can reason it out another way.

Answer 3

Do you know what dy/dt *means*?

Answer 4

y = (3n+1)/(n+1)
dy/dt = 2/(x+1)^2

Answer 5

ya i do

Answer 6

Okay then, so if limt->infy should give y(t) = 3, then, at infinity, what should the slope of y(t) be?

Answer 7

3?

Answer 8

If, at infinity, our function needs to keep getting closer to y=3, then what value should the SLOPE of the function be approaching?

Answer 9

infinity

Answer 10

wait no 0

Answer 11

what is the limit of y = (3n+1)/(n+1) at infinity?

Answer 12

@v4xN0s Yes, 0. What's another name for slope?

Answer 13

derivative

Answer 14

Yes, or, more relavant to this problem, dy/dt. Now, as you said, we want dy/dt = 0 when y = 3, right?

Answer 15

yep

Answer 16

Okay then. In ay+b, if I make y = 3, then what can a and b be?

Answer 17

so it would be y'=3-y?

Answer 18

That's one solution, b = 3 and a = -1.

Answer 19

no we want y = 3 when t = infinity?

Answer 20

@zzr0ck3r yes thats what the question says

Answer 21

what if all the solutions diverged from y=2

Answer 22

"diverged from y = 2" --> what?

Answer 23

as t approaches infinity

Answer 24

@v4xN0s we're not done with the previous one yet

Answer 25

oh alright i thought that was the write answer QQ

Answer 26

QQ? Anyway, that's just ONE anwer. In fact, any a and b that makes 3a+b = 0 works.

Answer 27

But it depends on the initial conditions, too.

Answer 28

oh i see well i just needed one diff eq for the problem

Answer 29

Okay, your example of dy/dt=3-y was a bit of a lucky guess.

Answer 30

See, if your initial condition is\[y_0>3\]then your slope is negative, and y decreases until it 'reaches' 3, at which point it 'stays' constant.

Answer 31

i see and if its less than 3 then its positive and increases as it gets closer to 3

Answer 32

Yes. If\[y_0=3\], then your solution is a line, starting at y=3 and staying that way (dy/dt = 0).

Answer 33

wait no slope still decreases

Answer 34

Um no.....