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Asylum15

  • 2 years ago

Differentiation Question! :(

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  1. Asylum15
    • 2 years ago
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    If y = \[5e ^{-4t} - 3e ^{-2t}\] show that \[\frac{ d^2y }{ dt^2 } + 6\frac{ dy }{ dt } +8y = 0\]

  2. Asylum15
    • 2 years ago
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    Derive Y?

  3. mark_o.
    • 2 years ago
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    use the formula D(e^u)= e^u D(u) lets see what will you get

  4. Asylum15
    • 2 years ago
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    So Y' = -4(5e^-4t) + 2 (-3e^-2t) ?

  5. Asylum15
    • 2 years ago
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    e^ax = ae^ax

  6. mark_o.
    • 2 years ago
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    De^ax = ae^ax yup

  7. Asylum15
    • 2 years ago
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    So my first step is correct?

  8. mark_o.
    • 2 years ago
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    ah +3

  9. Asylum15
    • 2 years ago
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    \[= -4(5e ^{-4t}) + 2(-3e ^{-2t})\]

  10. Asylum15
    • 2 years ago
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    ?

  11. mark_o.
    • 2 years ago
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    sorry about that i dint solved it my self but im just guiding you to the right solution :D

  12. Asylum15
    • 2 years ago
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    So am i right so far?

  13. mark_o.
    • 2 years ago
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    yes correct :D

  14. Asylum15
    • 2 years ago
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    Ok, what should I do next? :)

  15. mark_o.
    • 2 years ago
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    now do the 2nd derivative

  16. Asylum15
    • 2 years ago
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    Y'' yeah?

  17. mark_o.
    • 2 years ago
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    yup but you can still simplify that one by multiplying the constant coefficients

  18. Asylum15
    • 2 years ago
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    -4 multipled by 5e ?

  19. Asylum15
    • 2 years ago
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    Why is the 5e on the first line a -?

  20. mark_o.
    • 2 years ago
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    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

  21. Asylum15
    • 2 years ago
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    If you have -3e^-2t and the -2 comes down, wont you have +2 multiplying by -3 = -6? :D

  22. mark_o.
    • 2 years ago
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    its -3(-2)=+6

  23. Asylum15
    • 2 years ago
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    Oh :D

  24. mark_o.
    • 2 years ago
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    but for +3(-2)=-6

  25. Asylum15
    • 2 years ago
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    Sorry, so now do the 2nd derivative of this?

  26. mark_o.
    • 2 years ago
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    yeah try that

  27. Asylum15
    • 2 years ago
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    Any ideas how to start it? Aren't we back where we started?

  28. Asylum15
    • 2 years ago
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    I need to get an a/b no?

  29. mark_o.
    • 2 years ago
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    if y'=-20e^-4t +6e^-2t what is y" ?

  30. mark_o.
    • 2 years ago
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    or for example if y=-20e^-4t +6e^-2t what is dy/dx?

  31. Asylum15
    • 2 years ago
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    I know i have to derive that. But not sure how to start

  32. Asylum15
    • 2 years ago
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    Do you use the same rule again?

  33. mark_o.
    • 2 years ago
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    do the same thing as you did on the first one

  34. Asylum15
    • 2 years ago
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    ah

  35. mark_o.
    • 2 years ago
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    yup the same rule applies to it

  36. Asylum15
    • 2 years ago
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    80e^-4t - 12e^-2t

  37. mark_o.
    • 2 years ago
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    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

  38. Asylum15
    • 2 years ago
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    ? Don't know what you mean

  39. mark_o.
    • 2 years ago
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    just plug in the one you just solve to the formula d^2y/dt^2 +6 dy/dt +8y =0 _________ +6(_______) +8(________) =0

  40. mark_o.
    • 2 years ago
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    _________ +6(_______) +8(________) =? solve it and see if you will get zero

  41. Asylum15
    • 2 years ago
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    But Don't i have a t?

  42. mark_o.
    • 2 years ago
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    its like 6 dy/dt 6( -20 e^-4t -3e^-2t ) multiply it

  43. Asylum15
    • 2 years ago
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    Sorry this is taking so long, I got A in my last 3 engineering differentiation exams, but just not getting this one

  44. Asylum15
    • 2 years ago
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    But how can you say it's equal to zero if you have T and Y still in the equation?

  45. mark_o.
    • 2 years ago
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    6 dy/dt =6( -20 e^-4t -3e^-2t )= -120 e^-4t -18 e^-2t

  46. mark_o.
    • 2 years ago
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    you will add or subtract like terms like 100 e^ 7t + 50 e^ 7t = 150 e^ 7t

  47. Asylum15
    • 2 years ago
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    But T has no value, no given value, so I can't say for example 80^-4t = 0!?

  48. Asylum15
    • 2 years ago
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    I don't know T! :P

  49. Asylum15
    • 2 years ago
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    Using your method I have : -40e^-4t + 24e^-2t + 8y = 0

  50. mark_o.
    • 2 years ago
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    yup it has no value its like this 10 e^ 7t - 10 e^ 7t = 0 e^ 7t=0

  51. mark_o.
    • 2 years ago
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    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 ?

  52. Asylum15
    • 2 years ago
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    -40e^-4t + 24e^-2t + 8y = 0

  53. Asylum15
    • 2 years ago
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    :D

  54. mark_o.
    • 2 years ago
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    d^2y/dt^2 +6 dy/dt + 8y =? (80e^-4t - 12e^-2t) +6(-20e^-4t +6e^-2t) + 8( 5e^-4t -3e^-2t)=?

  55. mark_o.
    • 2 years ago
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    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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