Thank you so much. I was deriving an equation for a PIAB in three dimensions (cartesian coordinates) for my pchem midterm tonight and this concept was honestly the hardest part.
Probably a little late now but anyways. It’s an ordinary differential equation in which case you are just looking for a X(x) that relates to it’s derivatives. In this situation it is not too hard to see that sin and cos are heavily related to their second derivatives. They are just the negatives, so if you plug in either sin or cos for X(x) you’ll see it works out however using just one isn’t the whole answer. Hence why he uses both sin and cos with an arbitrary constant “a”,”b” this allows for all solutions to be covered in the singular answer. The reason this works is because the sum of two solutions to an ODE is in itself a solution to the same ODE. That proof has to do with some linear algebra but I hope this helps.
@@jordanlaforce2370 Probablya to late now but anyways. You get the sin cos solution by taking the test function e^{lambda * x}. When solving for your constant you get a complex solution and use eulers identity which gives a cos and sin solution.
Im sorry, may i ask something? Why you choose -lambda^2 as a constany, which is the constant is negatif. Why you not choose a constant positif or constanta 0, please tell me why? Thx before
Because when you find the general solution of the two ODE's you have to find the roots of the equations by square rooting and if it's just lambda or k instead of lambda^2 or k^2 you end up with a more complicated square root problem. It just easier to work with k^2 or lambda^2, than k or lambda.
The constant can be named anything you want. In this case, I knew that eventually I wanted solutions like sin(kx) with k real. In practice doing it yourself, you would likely first name the constant "C" or something and then realize later that sqrt(-C) is what appears naturally in your solutions. So you would rename it then.
This video clarified a technique that was being used in a book, which I was trying for days to figure out how it worked. Thank you for sharing.
Great work, love your channel. I’m a 50 yo BSEE and you’ve helped me keep my gears greased. Thank you Sam! 🙏
awesome, that's great to hear!
Thank you so much. I was deriving an equation for a PIAB in three dimensions (cartesian coordinates) for my pchem midterm tonight and this concept was honestly the hardest part.
I enjoyed your explanation, thanks for the video Sam
Excellent explaination, first time this makes sense to me
Sam, you're a star. Do you know that. Thanks a lot for you eloquence in explaining this.
Hey king you dropped this 👑
Thanks a lot cleary explained it!! Really awesome
Griffiths QM chapter 2.1, A man of culture I see
you just saved me from headache.
Amazing, thanks!
Great video, thankyou. I'm a bit confused on how you got the final equations, at 8:10 onwards though?
Probably a little late now but anyways. It’s an ordinary differential equation in which case you are just looking for a X(x) that relates to it’s derivatives. In this situation it is not too hard to see that sin and cos are heavily related to their second derivatives. They are just the negatives, so if you plug in either sin or cos for X(x) you’ll see it works out however using just one isn’t the whole answer. Hence why he uses both sin and cos with an arbitrary constant “a”,”b” this allows for all solutions to be covered in the singular answer. The reason this works is because the sum of two solutions to an ODE is in itself a solution to the same ODE. That proof has to do with some linear algebra but I hope this helps.
@@jordanlaforce2370 Probablya to late now but anyways. You get the sin cos solution by taking the test function e^{lambda * x}. When solving for your constant you get a complex solution and use eulers identity which gives a cos and sin solution.
Amazing video
Awesome man !
Great
5:36 wow thank yoou so much! Now I got it
So glad it was helpful, thanks!
Thanks a lot. really.
Sam please 🙏 upload more videos
how would you do non-homogenic?
also
how would you do non-separable?
Im sorry, may i ask something? Why you choose -lambda^2 as a constany, which is the constant is negatif. Why you not choose a constant positif or constanta 0, please tell me why? Thx before
Because when you find the general solution of the two ODE's you have to find the roots of the equations by square rooting and if it's just lambda or k instead of lambda^2 or k^2 you end up with a more complicated square root problem. It just easier to work with k^2 or lambda^2, than k or lambda.
@@j.pesquera Can we use lambda as a constant too?
@@diegofutgol87 Yes, lambda is a constant.
May god bless you
Thanks a lot
a boon. thanks mate
why did you say that the constant was -k^2 ?
The constant can be named anything you want. In this case, I knew that eventually I wanted solutions like sin(kx) with k real. In practice doing it yourself, you would likely first name the constant "C" or something and then realize later that sqrt(-C) is what appears naturally in your solutions. So you would rename it then.
@@SamGralla ah ok thank you!
You're 👍