Question

Differentiation Question! :(

Answer 1

If y = \[5e ^{-4t} - 3e ^{-2t}\] show that \[\frac{ d^2y }{ dt^2 } + 6\frac{ dy }{ dt } +8y = 0\]

Answer 2

Derive Y?

Answer 3

use the formula
D(e^u)= e^u D(u) lets see what will you get

Answer 4

So Y' = -4(5e^-4t) + 2 (-3e^-2t) ?

Answer 5

e^ax = ae^ax

Answer 6

De^ax = ae^ax yup

Answer 7

So my first step is correct?

Answer 8

ah +3

Answer 9

\[= -4(5e ^{-4t}) + 2(-3e ^{-2t})\]

Answer 10

?

Answer 11

sorry about that i dint solved it my self but im just guiding you to the right solution :D

Answer 12

So am i right so far?

Answer 13

yes correct :D

Answer 14

Ok, what should I do next? :)

Answer 15

now do the 2nd derivative

Answer 16

Y'' yeah?

Answer 17

yup but you can still simplify that one by multiplying the constant coefficients

Answer 18

-4 multipled by 5e ?

Answer 19

Why is the 5e on the first line a -?

Answer 20

hmm do it like this
5e^-4t -3e^-2t
=5e^-4t D(-4t) -3e^-2tD-2t)
=5(-4)e^-4t -3(-2)e^-2t
=-20e^-4t +6e^-2t

Answer 21

If you have -3e^-2t and the -2 comes down, wont you have +2 multiplying by -3 = -6? :D

Answer 22

its -3(-2)=+6

Answer 23

Oh :D

Answer 24

but for +3(-2)=-6

Answer 25

Sorry, so now do the 2nd derivative of this?

Answer 26

yeah try that

Answer 27

Any ideas how to start it? Aren't we back where we started?

Answer 28

I need to get an a/b no?

Answer 29

if
y'=-20e^-4t +6e^-2t
what is y" ?

Answer 30

or for example if
y=-20e^-4t +6e^-2t
what is dy/dx?

Answer 31

I know i have to derive that.

But not sure how to start

Answer 32

Do you use the same rule again?

Answer 33

do the same thing as you did on the first one

Answer 34

ah

Answer 35

yup the same rule applies to it

Answer 36

80e^-4t - 12e^-2t

Answer 37

yeah correct :D

now do some multiplication and addition or subtraction using the formula
to prove
the one you wrote down above
the d^2y/dx^2

Answer 38

? Don't know what you mean

Answer 39

just plug in the one you just solve to the formula

d^2y/dt^2 +6 dy/dt +8y =0

_________ +6(_______) +8(________) =0

Answer 40

_________ +6(_______) +8(________) =? solve it and see if you will get zero

Answer 41

But Don't i have a t?

Answer 42

its like 6 dy/dt
6( -20 e^-4t -3e^-2t ) multiply it

Answer 43

Sorry this is taking so long, I got A in my last 3 engineering differentiation exams, but just not getting this one

Answer 44

But how can you say it's equal to zero if you have T and Y still in the equation?

Answer 45

6 dy/dt =6( -20 e^-4t -3e^-2t )= -120 e^-4t -18 e^-2t

Answer 46

you will add or subtract like terms like 100 e^ 7t + 50 e^ 7t = 150 e^ 7t

Answer 47

But T has no value, no given value, so I can't say for example 80^-4t = 0!?

Answer 48

I don't know T! :P

Answer 49

Using your method I have : -40e^-4t + 24e^-2t + 8y = 0

Answer 50

yup it has no value its like this 10 e^ 7t - 10 e^ 7t = 0 e^ 7t=0

Answer 51

just plug in the one you just solve to the formula

d^2y/dt^2 +6 dy/dt + 8y =?
I I I
V V V <------- arrow down pointing it solve them
(_____________)+ 6(_________) + 8(_______)= __________ 0 ?

Answer 52

-40e^-4t + 24e^-2t + 8y = 0

Answer 53

:D

Answer 54

d^2y/dt^2 +6 dy/dt + 8y =?

(80e^-4t - 12e^-2t) +6(-20e^-4t +6e^-2t) + 8( 5e^-4t -3e^-2t)=?

Answer 55

d^2y/dt^2 + 6 dy/dt + 8y =?

(80e^-4t - 12e^-2t) + 6(-20e^-4t +6e^-2t) + 8( 5e^-4t -3e^-2t)=? solve by

multiplying first the 6 times the value of dy/dt also 8 times the value of y then add or

subtract like terms