Question

Solve the following equation using the Bernoulli substitution method:
y' = εy - δy³, ε>0 & δ>0

Answer 1

I got it down to 2v' + εv = δ but I'm not sure if that's correct or not.

Answer 2

yep

Answer 3

it's correct

Answer 4

So I found the integrating factor to be E^(εx) which leads the equation to
E^(εx)∗v=∫(E^(εx))∗δdx

delta can't be a constant right? So I would need to do a partial integration but this doesn't seem right...

Answer 5

\[e ^{\int\limits_{}^{}\epsilon } ?\]

Answer 6

aren't delta and epsilon supposed to be functions of x?

Answer 7

or just constants?

Answer 8

I'm not sure at this point, the constraints for the problem is delta >0 and epsilon > 0

:\

Answer 9

constants I guess... weird.. if that's so then you're right up to...

Answer 10

let me check

Answer 11

should have:

\[v = e^{-\epsilon x} * \int\limits_{ }^{ } e^{\epsilon x} * \delta * dx\]

Answer 12

yeah
you got that

Answer 13

simplifies nicely

Answer 14

\[\delta / \epsilon ?\]

Answer 15

for v? is that what you got?

Answer 16

I got... (δ/ϵ) + (C/e^(δx))

Answer 17

right good call, makes no sense otherwise (y would be a constant lol)

Answer 18

forgot my C

Answer 19

Ah alright, after some simplifying I got it to match the answer

Answer 20

\[y = v ^{-1/2}\]

so you're done

Answer 21

Thanks for your guidance :)

Answer 22

Thank you for yours, haven't played with these for years; didn't think anyone taught them anymore..

Answer 23

Heh, yeah I'm taking O.D.E. right now and the professor is going at a breakneck pace haha

Answer 24

good times... :) gl2u!

Answer 25

Thanks :p