Question

Solve the differential Equation dy/dx = (8e^x - 9e^-x)^2

Answer 1

please 8^x or x^8 ????

Answer 2

err crap (8e^x - 9e^-x)^2

Answer 3

2(8e^x-9e^-x)^1*8xe^x+9xe^-x

Answer 4

@aliahmed
You differentiated, but I think we needed to integrate :)

Answer 5

why?

Answer 6

...so expand, separate the variables, and integrate

Answer 7

because we want y and are given dy/dx

Answer 8

The entire equation was equal to dy/dx, so we have to reverse the process, don't we?

Answer 9

sorry, not equation, but expression, my bad :)

Answer 10

yeye we want y

Answer 11

Apparently for (8e^x - 4e^-x)^2 is 32e^(2x) - 64x -8^(-2x) +c

Answer 12

is it a 9 or a 4 for the other coefficient?

Answer 13

it's a 9, but this was a practice question I was doing with a 4.

Answer 14

So when we expand 8^x * 9^-x does it become 63^-2x?

Answer 15

with the e's in there too :P

Answer 16

Why did you multiply them? I thought they were to be subtracted?

Answer 17

there is no x that comes out

Answer 18

(8e^x - 9e^-x)^2 wouldnt' we do (8e^x - 9e^-x) (8e^x - 9e^-x)????

Answer 19

and then cross multiply?

Answer 20

well, expand (8e^x - 9e^-x)^2, whatever you call expanding it

Answer 21

it would be what you perhaps know as FOIL ?

Answer 22

Yes :)

Answer 23

yes

Answer 24

so FOIL out the expression above

Answer 25

...that is the same thing as "expanding" (8e^x-9e^-x)^2

Answer 26

64e^2x - 72e^-2x + 81e^2x?

Answer 27

not quite; careful with those exponents...

Answer 28

Not to mention you have to take the product of the two terms twice

Answer 29

yeah I was konfused with the exponents.

Answer 30

what are the terms?
look at the outer term
it will be
-(8e^x)(9e^-x)
can you simplify that?

Answer 31

By terms I meant 8e^x and 9e^-x
^^

Answer 32

oh woopsies I forgot to do the middle terms twice :P. Would it be -72e^-2x?

Answer 33

so 2x that would be 144e^4x?

Answer 34

no, check your exponents again
8e^x
9e^-x
:)

Answer 35

your law of exponents is\[x^a\cdot x^b=x^{a+b}\]

Answer 36

Idk I suck at exponents! They are supposed to be multiplied right ?? X * -X?

Answer 37

oh okay I was thinking that too.. SO it wouldbe x-x? so 0?

Answer 38

yes, because they have the same base we add exponents when we multply

Answer 39

and the x+(-x)=0
so what is \[\large 9e^x\cdot8e^{-x}\]?

Answer 40

well e^0 =i 1 so would it just be 72?

Answer 41

yes
and the other term?

Answer 42

ugh where did that 'i' come from haha.

Answer 43

I didn't even see it, lol

Answer 44

so it wouldbe -72-72 so -144?

Answer 45

:P

Answer 46

yep :)

Answer 47

so then we get 64e^2x - 144 + 81^-2x?

Answer 48

correct, and now we integrate

Answer 49

rather pause that
let's look at the process if it's new to you

Answer 50

so far we have expanded the RHS to give\[\frac{dy}{dx}=64e^{2x}-144+81^{-2x}\]now we "separate the variables
that is, we treat the symbol on the left as a fraction (though really, it isn't) and get the dx on the side with the x's like we would in algebra)

Answer 51

\[\frac{dy}{dx}=64e^{2x}-144+81^{-2x}\]\[dy=64e^{2x}-144+81^{-2x}dx\]now we can integrate both sides\[\int dy=\int64e^{2x}-144+81^{-2x}dx\]what do you get?

Answer 52

well e^u is still e^u so those exponents stay the same.

Answer 53

but according to the practice version is seems as though you divide by 2

so would it be 32e^2x - 144 - 81^-2x/2?

Answer 54

I guess that's because our u = 2x. But actually in that case it should be +(81^-2x)/2

Answer 55

hmm guess that's not right./... It also says in the practice version answer that the middle term is *x? so 144x? Idk if that makes sense... :(

Answer 56

Try doing it yourself, try to forget that you ever saw that practice version :)

Answer 57

I did as you can see above, but it's not right....

Answer 58

Well, if you think it's not right, let's just start it again :
\[\int\limits_{}^{}(64e^{2x} + 81e^{-x} - 144) dx\]
How do you go about solving this integral?

Answer 59

\[\int\limits_{}^{}(64e^{2x} + 81e^{-2x} - 144) dx\] ;)

Answer 60

^ yes lol

Answer 61

Well we can split it up into 3 integrals. u = 2x for the first, and u = -2x for the next.

Answer 62

it should come out to 32e^2x -81e^2x/2 - 144?

Answer 63

which is what I posted before...

Answer 64

or if u is just 2x then it would be +81e^2x/2

Answer 65

you forgot integrate -144

Answer 66

oh yeah so 144x

Answer 67

Ok we win :). yay!

Answer 68

I always forget stupid stuff like that... Thanks for your help as always Turing!

Answer 69

always welcome :)