Asylum15
  • Asylum15
Differentiation Question! :(
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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Asylum15
  • Asylum15
If y = \[5e ^{-4t} - 3e ^{-2t}\] show that \[\frac{ d^2y }{ dt^2 } + 6\frac{ dy }{ dt } +8y = 0\]
Asylum15
  • Asylum15
Derive Y?
anonymous
  • anonymous
use the formula D(e^u)= e^u D(u) lets see what will you get

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Asylum15
  • Asylum15
So Y' = -4(5e^-4t) + 2 (-3e^-2t) ?
Asylum15
  • Asylum15
e^ax = ae^ax
anonymous
  • anonymous
De^ax = ae^ax yup
Asylum15
  • Asylum15
So my first step is correct?
anonymous
  • anonymous
ah +3
Asylum15
  • Asylum15
\[= -4(5e ^{-4t}) + 2(-3e ^{-2t})\]
Asylum15
  • Asylum15
?
anonymous
  • anonymous
sorry about that i dint solved it my self but im just guiding you to the right solution :D
Asylum15
  • Asylum15
So am i right so far?
anonymous
  • anonymous
yes correct :D
Asylum15
  • Asylum15
Ok, what should I do next? :)
anonymous
  • anonymous
now do the 2nd derivative
Asylum15
  • Asylum15
Y'' yeah?
anonymous
  • anonymous
yup but you can still simplify that one by multiplying the constant coefficients
Asylum15
  • Asylum15
-4 multipled by 5e ?
Asylum15
  • Asylum15
Why is the 5e on the first line a -?
anonymous
  • anonymous
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
Asylum15
  • Asylum15
If you have -3e^-2t and the -2 comes down, wont you have +2 multiplying by -3 = -6? :D
anonymous
  • anonymous
its -3(-2)=+6
Asylum15
  • Asylum15
Oh :D
anonymous
  • anonymous
but for +3(-2)=-6
Asylum15
  • Asylum15
Sorry, so now do the 2nd derivative of this?
anonymous
  • anonymous
yeah try that
Asylum15
  • Asylum15
Any ideas how to start it? Aren't we back where we started?
Asylum15
  • Asylum15
I need to get an a/b no?
anonymous
  • anonymous
if y'=-20e^-4t +6e^-2t what is y" ?
anonymous
  • anonymous
or for example if y=-20e^-4t +6e^-2t what is dy/dx?
Asylum15
  • Asylum15
I know i have to derive that. But not sure how to start
Asylum15
  • Asylum15
Do you use the same rule again?
anonymous
  • anonymous
do the same thing as you did on the first one
Asylum15
  • Asylum15
ah
anonymous
  • anonymous
yup the same rule applies to it
Asylum15
  • Asylum15
80e^-4t - 12e^-2t
anonymous
  • anonymous
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
Asylum15
  • Asylum15
? Don't know what you mean
anonymous
  • anonymous
just plug in the one you just solve to the formula d^2y/dt^2 +6 dy/dt +8y =0 _________ +6(_______) +8(________) =0
anonymous
  • anonymous
_________ +6(_______) +8(________) =? solve it and see if you will get zero
Asylum15
  • Asylum15
But Don't i have a t?
anonymous
  • anonymous
its like 6 dy/dt 6( -20 e^-4t -3e^-2t ) multiply it
Asylum15
  • Asylum15
Sorry this is taking so long, I got A in my last 3 engineering differentiation exams, but just not getting this one
Asylum15
  • Asylum15
But how can you say it's equal to zero if you have T and Y still in the equation?
anonymous
  • anonymous
6 dy/dt =6( -20 e^-4t -3e^-2t )= -120 e^-4t -18 e^-2t
anonymous
  • anonymous
you will add or subtract like terms like 100 e^ 7t + 50 e^ 7t = 150 e^ 7t
Asylum15
  • Asylum15
But T has no value, no given value, so I can't say for example 80^-4t = 0!?
Asylum15
  • Asylum15
I don't know T! :P
Asylum15
  • Asylum15
Using your method I have : -40e^-4t + 24e^-2t + 8y = 0
anonymous
  • anonymous
yup it has no value its like this 10 e^ 7t - 10 e^ 7t = 0 e^ 7t=0
anonymous
  • anonymous
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 ?
Asylum15
  • Asylum15
-40e^-4t + 24e^-2t + 8y = 0
Asylum15
  • Asylum15
:D
anonymous
  • anonymous
d^2y/dt^2 +6 dy/dt + 8y =? (80e^-4t - 12e^-2t) +6(-20e^-4t +6e^-2t) + 8( 5e^-4t -3e^-2t)=?
anonymous
  • anonymous
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

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