Question

\[\frac{dy}{dx}-y=6\] is linear, you can look up what that means

Answer 1

ok

Answer 2

idea is this

Answer 3

you want to make the left hand side look like the derivative of a product, my multiplying by the "integrating factor"

Answer 4

Alright

Answer 5

in other words you find something, the book calls it \(\mu\) so that when you multiply you will have \[\mu \frac{dy}{dx}-\mu y=6\mu\] adn \(\mu=\mu(x)\) a function of x

Answer 6

damn typo

Answer 7

i'll get this eventually

Answer 8

example in the book is \[\frac{dy}{dx}-3y=6\]

Answer 9

so now how to find \(\mu\) looks complicated but it is easy

Answer 10

Looks tough honestly

Answer 11

first we have \(-3\) in front of the y
step one is to integrate wrt x i.e .\[\int-3dx\]

Answer 12

let me know what you get

Answer 13

Ok -3x

Answer 14

right
step two is take \(e^{-3x}\) and we rare done, that is \(\mu\)

Answer 15

Why?

Answer 16

And we are looking for mu?

Answer 17

lets see why

Answer 18

we found \(\mu\) it is \(e^{-3x}\)

Answer 19

Thats it?

Answer 20

ok now lets multiply and see what magic happens

Answer 21

oh no, that i just \(\mu\) we are not done
lets multiply both sides by \(e^{-3x}\)

Answer 22

\[\frac{dy}{dx}-3y=6\]
\[e^{-3x}\frac{dy}{dx}-3e^{-3x}y=6e^{-3x}\]

Answer 23

now look at the left hand side and see what you have
why, it is the derivative of \(e^{-3x}y\) a miracle
check it

Answer 24

one sec brb

Answer 25

by the produce rule \[\frac{d}{dx}[e^{-3x}y]=e^{-3x}\frac{dy}{dx}-3e^{-3x}y\]

Answer 26

back

Answer 27

Ok ?

Answer 28

ok

Answer 29

so is it clear that the left hand side is a derivative, the derivative of \[e^{-3x}y\]?

Answer 30

yes

Answer 31

then to solve this \[e^{-3x}\frac{dy}{dx}-3e^{-3x}y=6e^{-3x}\] integrating you get \[e^{-3x}y=\int 6e^{-3x}dx+c\]

Answer 32

let me know when that is clear

Answer 33

So -2e^(-3x) + c?

Answer 34

final answer \[e^{-3x}y=-2e^{-3x}+c\]

Answer 35

Okie

Answer 36

actually final answer is best to solve for y

Answer 37

So divide by e^(-3x)

Answer 38

multiply both sides by \(e^{-3x}\) get \[y=-2+ce^{3x}\]

Answer 39

Oh right right

Answer 40

oops i meant multiply both sides by \(e^{3x}\)

Answer 41

check that is works

Answer 42

want to try one more?

Answer 43

Please

Answer 44

ok let me pick one
first step is to find \(\mu\) we can do say 5

Answer 45

Okie dokie