## Astrophysics one year ago Leibniz notation product rule... @ganeshie8 just be expected to be tagged

1. Astrophysics

$\frac{ d^2y }{ dx^2 }$ I'm trying to get this to equal $\frac{ d^2y }{ dt^2 }t^2+\frac{ dy }{ dt }t$ so I have $\frac{ d^2y }{ dx^2 } = \frac{ d }{ dx }\left( \frac{ dy }{ dt } t\right)$ which I got from the chain rule but how do I use the product rule here X_X

2. Astrophysics

|dw:1444614858695:dw|

3. Astrophysics

@freckles

4. Kainui

Wait so are we saying y(x) and x(t) is that true?

5. Astrophysics

Well I'm trying to figure out eulers equation $t^2\frac{ d^2y }{ dt^2 }+\alpha t \frac{ dy }{ dt }+ \beta y=0~~~t>0$ and it says I need to calculate dy/dt and d^2y/dt^2 in terms of dy/dx and d^2y/dx^2

6. Astrophysics

so I can use my substitution

7. Kainui

Oh ok this makes more sense now.

8. Kainui

wait are you not just allowed to substitute in $y=At^n$ and solve?

9. Astrophysics

No I have to use x = lnt

10. Astrophysics

and it's telling me to do all that stuff lol

11. Kainui

Oh alright then

12. Astrophysics

Just want to know how to use the product rule in leibniz notation dont really care about the problem

13. Astrophysics

f'g+g'f lol but I can't believe I don't know how to do this

14. Kainui

$y(x(t))$ $\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt}$ $\frac{d}{dt} \left( \frac{dy}{dx} \frac{dx}{dt} \right) = \frac{d}{dt} \left( \frac{dy}{dx} \right)\frac{dx}{dt} +\frac{dy}{dx}\frac{d}{dt} \left( \frac{dx}{dt} \right)$ Try to take it from here.

15. Kainui

While you do that, I'll do it without logs to see how much faster it is.

16. Astrophysics

It's the first part, $\frac{ d }{ dt }\left( \frac{ dy }{ dx } \right)\frac{ dx }{ dt }$ there's no changeeee, I was expecting there to be d^2y/dx^2 or something

17. Kainui

Ultimately they'll get the same answer and this method you're using is more general so it's not without its merits it's just this is a pretty common one to solve.

18. Kainui

Here's only the derivative part of the first part (notice we'll have a squared dx/dt when we multiply this back in): $\frac{d}{dt} \left( \frac{dy}{dx} \right) = \frac{d^2y}{dx^2} \frac{dx}{dt}$

19. Astrophysics

Ok should it not be $\frac{ d^2y }{ dx^2 }\frac{ d }{ dt }$

20. Kainui

If this bothers you instead let: $\frac{dy}{dx} = u(x(t))$ Then when you do the derivative you have: $\frac{d}{dt}(u) = \frac{du}{dx} \frac{dx}{dt}$ Then you can look at the du/dx term on its own as the second derivative of y wrt x.

21. Kainui

Explain your reasoning for this: $\frac{ d^2y }{ dx^2 }\frac{ d }{ dt }$

22. Astrophysics

Nvm I'm dumb, it's $\frac{ d }{ dt }\left( \frac{ dy }{ dx } \right) = \frac{ dy }{ dt }\frac{ dt }{ dx }\frac{ d }{ dt } = \frac{ d^2y }{ dt^2 }\frac{ dt }{ dx }$ is this what you mean

23. Astrophysics

and not dx^2

24. Kainui

Nope, you're taking the derivatives in a strange order, and this hanging d/dt that's by itself is kind of meaningless here.

25. Astrophysics

I just used the chain rule, the d/dt is what is confusing me

26. Kainui

$y(x(t))$ $\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt}$ $\frac{d}{dt} \left( \frac{dy}{dx} \frac{dx}{dt} \right) = \color{red}{ \frac{d}{dt} \left( \frac{dy}{dx} \right)}\frac{dx}{dt} +\frac{dy}{dx}\frac{d}{dt} \left( \frac{dx}{dt} \right)$ Looking only at the red part: $\frac{d}{dt} \left( \frac{dy}{dx} \right) = \frac{d^2y}{dx^2} \frac{dx}{dt}$ make the u substitution I described earlier to get through this part.

27. Kainui

If there's anything about using the product and chain rule above that confuses you say something now before moving forward.

28. Astrophysics

Nooo, it's not really the substitution part, I mean $\frac{d}{dt} \left( \frac{dy}{dx} \frac{dx}{dt} \right) = \color{red}{ \frac{d}{dt} \left( \frac{dy}{dx} \right)}\frac{dx}{dt} +\frac{dy}{dx}\frac{d}{dt} \left( \frac{dx}{dt} \right)$ it was this in general, so you're just writing $\frac{ d }{ dt }\left( \frac{ dy }{ dx } \frac{ dx }{ dt }\right)$ + this thing again, oh I'm overthinking this, I just realized how simple it actually is

29. Astrophysics

$\frac{ d }{ dx }\left( \frac{ dy }{ dt }t \right)=\frac{ d }{ dx }\left( \frac{ dy }{ dt } \right)t+\frac{ d }{ dx }\frac{ dy }{ dt } (t)$

30. Astrophysics

omggggggg kai

31. Kainui

Where did this come from?

32. Kainui

Explain why you're trying to use this: $\frac{ d }{ dx }\left( \frac{ dy }{ dt }t \right)=\frac{ d }{ dx }\left( \frac{ dy }{ dt } \right)t+\frac{ d }{ dx }\frac{ dy }{ dt } (t)$

33. Astrophysics

That's my original q

34. Kainui

Isn't this what you're trying to solve? $t^2\frac{ d^2y }{ dt^2 }+\alpha t \frac{ dy }{ dt }+ \beta y=0~~~t>0$

35. Astrophysics

$\frac{ d^2y }{ dx^2 } = \frac{ d }{ dx }\left( \frac{ dy }{ dt } t\right) = \frac{ d }{ dx }(\frac{ dy }{ dt })t+\frac{ d }{ dx }\left( t \right)\frac{ dy }{ dt }$

36. Astrophysics

I was trying to find d^y/dt^2 in terms of d^2y/dx^2 ahaha

37. Astrophysics

I think I got it, I sort of just thought about this $\frac{ d }{ dx }(u*v)=...$ I was just overthinking

38. Kainui

No stop saying that lol, you can't claim that until you've finished solving the problem xD Right now I'm trying to get us on the path of writing $$\frac{d^2y}{dt^2}$$ in terms of x with the chain rule, but I'm not sure where you're going wrong so I'm having difficulty helping you out.

39. Astrophysics

Oh I thought we were doing it in terms of d^2y/dt^2 the whole time

40. Kainui

I don't know what you're doing so I don't know

41. Astrophysics

lmao

42. Kainui

$\frac{ d^2y }{ dx^2 } = \frac{ d }{ dx }\left( \frac{ dy }{ dt } t\right)$ Explain where you're getting this from, cause I don't know. That doesn't mean it's wrong it just means you need to say words, I can't read your mind.

43. Astrophysics

All I was doing is finding $\frac{ d^2y }{ dt^2 }$ in terms of $\frac{ d^2y }{ dx^2 }$ so I would get $\frac{ d^2y }{ dt^2 }t^2+\frac{ dy }{ dt }t$

44. Kainui

Hmmm explain how you're supposed to solve this: " it says I need to calculate dy/dt and d^2y/dt^2 in terms of dy/dx and d^2y/dx^2" So I am not seeing this here, so for example, dy/dt in terms of dy/dx would look like this, right: $\frac{dy}{dt} = \frac{dy}{dx}\frac{dx}{dt}$

45. Astrophysics

$\frac{ d^2y }{ dx^2 }$ and started using the chain rule, $\frac{ d^2y }{ dx^2 } = \frac{ d }{ dx }\left( \frac{ dy }{ dx } \right)=\frac{ d }{ dx }\left( \frac{ dy }{ dt }\frac{ dt }{ dx } \right)=\frac{ d }{ dx }\left( \frac{ dy }{ dt } t\right)$ and then I was trying to apply chain rule to $\frac{ d }{ dx }\left( \frac{ dy }{ dt } t\right)$

46. Astrophysics

I started with $\frac{ d^2y }{ dx^2 }$

47. Astrophysics

product rule not chain rule* omg

48. Kainui

So you're writing $\frac{d^2y}{dx^2}$ in terms of $\frac{d^2y}{dt^2}$ then? And you're plugging in $\frac{dt}{dx}=t$ is that right?

49. Astrophysics

Yes!

50. Kainui

This is fine then I think, it's just weird cause instead of looking it as y(x(t)) you're looking at it as y(t(x)) which is kind of backwards but in this case works since $$x= \ln t$$ which is surjective.

51. Astrophysics

Haha, I thought that's how you were suppose to do it, I read it on pauls online notes or something, I really have no notes for this, so I was sort of coming up with stuff myself

52. Astrophysics

I mean the point of getting this is so I can set up my equation as $\frac{ d^2y }{ dx^2 }+(\alpha - 1)\frac{ dy }{ dx }+\beta y=0$

53. Kainui

Right of course, but doing it like this is awkward since time isn't generally thought of as a function of position. And it has no meaning in periodic motion.

54. Astrophysics

true

55. Astrophysics

I wasn't really thinking of it that way, haha just some math problem

56. Astrophysics

And the question stated, let x = lnt and calculate dy/dt and d^2y/dt^2 in terms of dy/dx and d^2y/dx^2 ahah

57. Kainui

Here, I'll just walk through how I did it: $\frac{dy}{dt} = \frac{dy}{dx}\frac{dx}{dt}$ $\frac{d^2y}{dt^2} = \frac{d^2y}{dx^2} \left( \frac{dx}{dt} \right)^2 + \frac{dy}{dt} \frac{d^2x}{dt^2}$ I guess we can do it now or later, so why not now: $\frac{dx}{dt} = \frac{1}{t}$ $\frac{d^2x}{dt^2} = \frac{-1}{t^2}$ And then plugging all this in you get the same simplified differential equation since the $$t^2$$ will disappear.

58. Kainui

I messed up one piece but, oh well should have been a dy/dx on that last term I just noticed

59. Kainui

At least this way is more straight forward to plug in (well in my opinion). However I wouldn't use this method at all lol. Best way is to notice you have a power next to the derivative which is basically compensating for a lost power by the power rule every derivative. Still pretty easy to spot. $t^2 y''+\alpha t y' + \beta y = 0$ Now you can just pick $y=At^n$ and plug it in: $n(n-1)+\alpha n + \beta = 0$ Now you have a quadratic in n to solve for the two solutions, done!

60. Astrophysics

Yo that's so simple

61. Astrophysics

Yeah I already hate this method I have to use

62. Kainui

Haha use the hate to fuel you through the tedious calculations and before you know it you'll be really fast. I feel like most of this discussion didn't have to happen I just had no idea what you were doing so we wasted a lot of time.

63. Astrophysics

Yeah haha, well at least I know what I'm doing now X_X

64. Astrophysics

I just realized how simple this was, I actually just misread something earlier, so was sort of confused haha xD