## Asylum15 2 years ago Differentiation Question! :(

1. Asylum15

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

2. Asylum15

Derive Y?

3. mark_o.

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

4. Asylum15

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

5. Asylum15

e^ax = ae^ax

6. mark_o.

De^ax = ae^ax yup

7. Asylum15

So my first step is correct?

8. mark_o.

ah +3

9. Asylum15

$= -4(5e ^{-4t}) + 2(-3e ^{-2t})$

10. Asylum15

?

11. mark_o.

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

12. Asylum15

So am i right so far?

13. mark_o.

yes correct :D

14. Asylum15

Ok, what should I do next? :)

15. mark_o.

now do the 2nd derivative

16. Asylum15

Y'' yeah?

17. mark_o.

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

18. Asylum15

-4 multipled by 5e ?

19. Asylum15

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

20. mark_o.

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

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

22. mark_o.

its -3(-2)=+6

23. Asylum15

Oh :D

24. mark_o.

but for +3(-2)=-6

25. Asylum15

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

26. mark_o.

yeah try that

27. Asylum15

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

28. Asylum15

I need to get an a/b no?

29. mark_o.

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

30. mark_o.

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

31. Asylum15

I know i have to derive that. But not sure how to start

32. Asylum15

Do you use the same rule again?

33. mark_o.

do the same thing as you did on the first one

34. Asylum15

ah

35. mark_o.

yup the same rule applies to it

36. Asylum15

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

37. mark_o.

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

? Don't know what you mean

39. mark_o.

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.

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

41. Asylum15

But Don't i have a t?

42. mark_o.

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

43. Asylum15

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

44. Asylum15

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

45. mark_o.

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

46. mark_o.

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

47. Asylum15

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

48. Asylum15

I don't know T! :P

49. Asylum15

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

50. mark_o.

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

51. mark_o.

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

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

53. Asylum15

:D

54. mark_o.

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.

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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