I'm a grad student in applied physics at Cornell, and these videos are the first time that someone has lectured math in a way that makes sense. This is life changing.
please please pleaseeee make video on generating function for legendry equation, hermite and its all equations.. humble request..please... ur videos are really helping me for my exams...
Just subscribed cause this helped me understand legendre eqn. I am doing this straight from high school too at a liberal arts college with not a lot of math support and engineering students and support
When you started to take the derivative of the power series at 1:49 you should start the summation from 1 for the first derivative and from 2 for the second one
I don't have to, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2 like you said, but I wanted to keep the starting point consistent.
No, it doesn't have to. It doesn't matter, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2, but I wanted to keep things consistent.
I'm an undergrad at a pretty reputed indian university but what wonders me is the fact that the faculty here despite pretty qualified and paid a handsome amount money don't possess even 0.0001% of teaching skills that you have. I literally skipped most of my math lectures because the professor's teaching style was such bogus that i lost interest in the subject. This video was a lifesaver for me.
hello! Great video as always. One question though. For k=1 the solution is in terms of odd coefficients and for k=2 in terms of even coefficients. So we say that the polynomial created by those even/odd coefficients is the solution. But, for k=1 we also have the even series that doesn't terminate. The even series for k=1 isn't an acceptable solution? We just disregard it ? Thanks in advance!
hi @Faculty of Khan , please enlighten me. when you assign a value for a0 & a1, is there an exact step to do so? coz if i pick different values. it will end up different formula. are there any ways to determine the exact value of a0 = -1/2 ?
using rodriguez formula: P2(x)= (1/2)(3x^2-1) but in legendre that you presented. a0 is arbitrary constant. that is my question. how to know the exact value of a0 which is -1/2
A fast-paced video which perfectly explains the math? Damn man, this is some next-level stuff. Bro, keep up the good work. I wish you my heartiest wish for your development of this channel.
for first and second derivative the starting value of n should change right ( 1 and 2) respectively , in order to make the function defined . But its not have been done , any specific reason ?
sir please why Ps are made different from y's n what determines the constants ? ok got it. so the function has value +/_ 1 at the ends of the x interval . ?
Ok can you help me out here? Why do we not consider the infinite series in the Legendre Polynomials? So e.g.: if k = 1, then why do we not consider the even series?
I think there is a little mistake if we put n equal to 0 in the first and second derivative of y(x) then it will be vanish so how they forthere proceeds I think n should be start from 1 in first derivative and n start from 2 in second derivative then we use. Power series properly if we use frobenius method then r must be required in the power of x in both first and second derivative and also in y(x)
No problem, thank you for the kind words! And yeah, that's just how I make my videos (faster writing just allows the lecture to flow better IMO): hopefully it's not too bad but you can still slow down the video or pause.
The even series is still part of the solution to the ODE. It's just that it's not a Legendre polynomial because it continues forever (doesn't terminate). Hope that helps!
Still a Legendre function, though, which is kinda "in between" the two subsequent Legendre polynomials, kinda like a fraction is in between two whole numbers ;)
Someone tell me where the x comes from in the formulas for y sub odd and y sub even after you do the recusion formula to find the constants. And how is P(x) the same thing as x?
That's just the series solution: the coefficient a_1 is attached to x, the coefficient a_2 is attached to x^2 and so on. And the P(x) you're referring to is actually P_1(x) (i.e. the first Legendre polynomial, which basically means the Legendre polynomial which has 1 as the highest power on x. Similarly, P_2(x) is the 2nd Legendre polynomial, so the Legendre polynomial with 2 as the highest power on x).
This is enlightening. Awesome explanation. This is what iv'e been stuck on for a while in quantum and EnM, because the last math class that's required in our physics major is calc 4 and maybe vector calc, haha so it just pops up out of no where a lot and it's hard picking up on some of these little things. Thanks so much for this, really really saved me a lot of scrambling and frustration.
Watching the video all the way from México. Thanks a lot for the great help, keep on with the greath Job. I have a doub't. What happen when I take k=n?
Thank you Gerardo! When k = n, the coefficient becomes zero, and the series terminates at that coefficient (giving you a polynomial solution). So for example, when you have k = 3, the a5 term becomes zero because of the (3-k) factor. I believe I discuss this in the video. Hope that helps!
A silly question but what's the difference between creating a dummy variable and shifting the index of the series? Are these just two methods that do the same thing?
Because the n = 0 term in the derivative (and the n = 0, 1 terms in the second derivative) is zero anyway. It's not necessary to change the index; I didn't do it so I could keep things simple and consistent.
hi.. thanks fr the video... I have a question.. The series we get After putting the value of y and its derivatives in the differential eqn. Is it necessary to make power of each sum consistent.. What if in some other case the values at m=-1 and m= - 2 doesn't evaluate to zero..
My question is this: How can we use a polynomial from one part of the solution (which converges) and disregard the other part that diverges? Should the two parts of the series be seen as a linear combination of solutions, and we just take the part of the solution that successfully solves the equation for the given value of l?
For anyone else who had the same question (I did), thought about this and came to this conclusion… Since y is a linear combo of odd and even, if the odd terminates but the even diverges, then a0 must be =0 in order for the series to converge and describe y. If the even terminates, then the odd won’t and a1 must be =0 to converge to y.
You probably could, but when you look at the indicial equation, your solution would just be r = 0, which is the same as the regular series solution method.
It doesn't matter, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2 like you said, but I wanted to keep things consistent.
Also, I know Legendre polynomials are orthogonal under the inner product that is the integral of the product from -1 to 1. Is there an easy way to tell this from the differential equation?
If one series terminate and another doesn't, then we still have infinte series representation since y is a combination of odd and even series, so the solution is still unsolved! What's wrong?
It's not unsolved; it's just that one of the solutions is a finite polynomial while the other is an infinite series. The finite polynomial is dubbed the Legendre polynomial.
I just used them so I could get expressions for the Legendre polynomials. For those exact values of a0, we have the Legendre polynomials. Hope that helps!
@@FacultyofKhan if we use different value of a0 or a1.. that will derived different answers. it will not be equal anymore using rodriguez formula. please enlighten us
There's a slight misleading information here. There is a normalization constant acting upon the solution of the 'y1' and 'y2'. You also have to mention them. The arbitrary constant doesn't make up the value.
P2(x) corresponds to k = 2, and y_even represents the terms that come from using even indices k on the recursion relation. Also, for a0 = -1/2, y_even becomes the second-order Legendre polynomial, which I've denoted as P2. Hope that helps!
the summation of series is zero ...so how can you justify it by saying that all the coefficients that are associated with the increasing power of the variable x is zero .... can it not be the case that terms in the series have a distribution of positive and negative values ?
legendre equation is used to obtain the potential due to gravitational system where there are no charges and masses . obtain the series solution to get potential
Well, I haven't made a video on that right now. But you can look here starting at section 2.2: www.physics.usu.edu/Wheeler/EM3600/Notes08SolvingForPotentials.pdf Hope that helps!
It doesn't matter if the summation is zero for m = 0 and m = 1; I could start it at m = 2 like you said but it doesn't make a difference since the m = 0 and m = 1 terms are zero anyway.
I have this doubt that when we shift the index of the first series to make x^s common then we do not shift the index of the rest of the equation. When we differentiate the series twice we are supposed to get A1 constant for first differentiation and A2 term for second. Then why haven't you showed it. Just for the sake of getting X^s common we cannot change the series according to our convenience.
I'm not sure I fully understand your question, but I'll try to explain it according to what I pieced together. I feel like you've asked 2 questions: 1. When you shift the index of the 1st series (around 3:00), why don't you shift the index of all the other series? Answer: The reason is that for now, I just want to make sure the x's are all raised to the same power, so it's easier to change the first series. 2. Why don't you change the starting value of n when you differentiate the series solution y = sum from n = 0 of an*x^n? Answer: I've discussed it in previous responses, but it doesn't matter, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2, but I wanted to keep things consistent.
Or start picking up lonely mechanical engineer girls ;) You can always impress them with solving some crazy ODE :) I wouldn't go with muscles myself, because chick that love muscles are usually empty and boring as hell :P so you don't want to hang out with them anyway, trust me :q
@@3631162 Relax and lighten up cringehere. So just because textbooks show photos of Laplace in formal coat and not laughing does not mean he did not have fun. Think about that.
lol well I can't really change my voice now; still, I hope you at least found the lesson useful. You can also turn on captions and put it on mute if the voice is too boring.
Sure! If you go back to 8:06, you can see the recursion relation at the top (a_n+2 as a function of a_n). All the last part is saying is that when k is an ODD number, the coefficients a_n for ODD values of n continue only up till a certain point, because when n reaches k, you can see that the numerator of the recursion relation immediately becomes zero. Now if one of the coefficients (a_n or a_k) reaches zero, all coefficients that come after that (e.g. a_n+2, a_n+4, etc) for odd values of n will also be zero. This means that the 'odd' index series would terminate, and we end up with a polynomial involving odd powers of x (e.g. P_1, P_3 etc). The same logic applies to EVEN values of k, in which the coefficients a_n for EVEN values of n eventually terminate. If you have any more questions, please ask!
I'm a grad student in applied physics at Cornell, and these videos are the first time that someone has lectured math in a way that makes sense. This is life changing.
Thank you so much!
Any tips on how hard was to get into grad school at Cornell?
OMG a fellow Cornelian!
i guess I am quite off topic but does anybody know of a good website to watch new movies online?
@@finleyahmir7054 😬
You're a godsend, you're literally the reason I'm passing Advanced Math for Physics this semester, thanks a lot!
Thank you! I'm glad you find my lectures useful!
Found this 10 Hrs before the quiz. Thank You!
Glad you found it useful!
Mine's in like an hour
please please pleaseeee
make video on generating function for legendry equation,
hermite and its all equations..
humble request..please...
ur videos are really helping me for my exams...
I'll put it on my to-do list.
You rock man!
BTW speed is unnaturally fast which suits the mathematical things you say...
Keep on doing good work, I will get a good grade in maths
Just subscribed cause this helped me understand legendre eqn. I am doing this straight from high school too at a liberal arts college with not a lot of math support and engineering students and support
Welcome!
thank you so much!! i am studying nuclear reactor analysis and this video saved my life.
Hopefully it will also save the lives of all the people who could have been blown up by a nuclear reactor meltdown if this video didn't help you ;)
@@scitwi9164 😂
you are the person who will be responsible for me passing mm3
Haha thank you! Though I'm curious what mm3 is.
mathematical methods 3, its year 2 physics course at ucl (university college london)
That sounds like an interesting course. Good luck!
Danny Best its as if this guy made these videos for us, small world
your way of teaching is good .
Glad you like it! Thanks!
I love the hint of comedy in your videos. Thank you for these videos. They are so clear and easy to understand
When you started to take the derivative of the power series at 1:49 you should start the summation from 1 for the first derivative and from 2 for the second one
I don't have to, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2 like you said, but I wanted to keep the starting point consistent.
Can someone tell me which software he is using for writing purpose??
Shouldn't the index changes as you differentiate?
No, it doesn't have to. It doesn't matter, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2, but I wanted to keep things consistent.
@@FacultyofKhan bumping the indexes at least for the second derivative would save us some headache later with matching the indexes ;)
I'm an undergrad at a pretty reputed indian university but what wonders me is the fact that the faculty here despite pretty qualified and paid a handsome amount money don't possess even 0.0001% of teaching skills that you have. I literally skipped most of my math lectures because the professor's teaching style was such bogus that i lost interest in the subject. This video was a lifesaver for me.
Once again, you nailed it man!
This video is really helpful. Keep on making more videos!
Good handwriting I like it haha
Thank you! Glad you liked it!
hello! Great video as always. One question though. For k=1 the solution is in terms of odd coefficients and for k=2 in terms of even coefficients. So we say that the polynomial created by those even/odd coefficients is the solution. But, for k=1 we also have the even series that doesn't terminate. The even series for k=1 isn't an acceptable solution?
We just disregard it ? Thanks in advance!
Amazingly Explained!
Thank you! Glad you liked it!
Very nice playlist explained in a concise way! thanks alot
hi @Faculty of Khan , please enlighten me. when you assign a value for a0 & a1, is there an exact step to do so? coz if i pick different values. it will end up different formula. are there any ways to determine the exact value of a0 = -1/2 ?
using rodriguez formula: P2(x)= (1/2)(3x^2-1)
but in legendre that you presented. a0 is arbitrary constant. that is my question. how to know the exact value of a0 which is -1/2
@@adonisbibat3895 I have this exact same question right now. Did you figure it out?
@@randytudor418 I'm confused about it too. would anyone please enlighten me please?
A fast-paced video which perfectly explains the math? Damn man, this is some next-level stuff. Bro, keep up the good work. I wish you my heartiest wish for your development of this channel.
Superb... techniques to show mathematical rigorous.
why are you not starting sums of y' from n=1 to infinity and y'' from n=2 to infinity because n is 0 at n=0 and (n-1)n is 0 when n=0,1?
10:19 just leaving a comment because I'm a special person
for first and second derivative the starting value of n should change right ( 1 and 2) respectively , in order to make the function defined . But its not have been done , any specific reason ?
Very well explained. Thanks a lot!
sir please why Ps are made different from y's n what determines the constants ?
ok got it. so the function has value +/_ 1 at the ends of the x interval . ?
Hey! Very clear and concise. Kudos.
If you had time, and don't mind, could you do a video on how you would derive the rodrigues formula via recurrsion relation. Thank you!
Sure thing! I'll put that on my to-do list. It shouldn't be too hard considering that I have the material already.
I'll also link it here so you can see when I put it up. Expect another reply in a week or so.
Thanks a lot, that would be great!!! ^.^
Ok can you help me out here? Why do we not consider the infinite series in the Legendre Polynomials? So e.g.: if k = 1, then why do we not consider the even series?
Thank you so much for your explanation, one question, which device &app you used for digital handwriting
I think there is a little mistake if we put n equal to 0 in the first and second derivative of y(x) then it will be vanish so how they forthere proceeds I think n should be start from 1 in first derivative and n start from 2 in second derivative then we use. Power series properly if we use frobenius method then r must be required in the power of x in both first and second derivative and also in y(x)
Thank you so much for these videos ...
Could you do one on Spherical Harmonics ?
Thank you :D
No problem! I'll definitely put that on my to-do list. My schedule this term is rather packed but whenever I get time, I'll get to that to-do list!
Nope. Graduate student :)
Awesome.. Good luck.. :D
thanks for explaining the concepts!
you wrote super fast
No problem, thank you for the kind words! And yeah, that's just how I make my videos (faster writing just allows the lecture to flow better IMO): hopefully it's not too bad but you can still slow down the video or pause.
your videos are so easy to understand...i really loved those but I would request you for power series solution of Laguerre differential equation
No problem! I'll definitely put that on my to-do list. Thank you for the feedback!
But sir can you have the orthonormality of polynomials also
Sure, I'll try.
but sir as soon as possible please.. because my finals are approaching and I have to prepare
Can I find a derivation legendre polynomials
wonderful, amazing explanation
how did you find a0-3a0*x^2???
For k ＝1， the even series does not terminate, so that's the reason why we do not take it for our solution of Legendre's equation of k=1?
The even series is still part of the solution to the ODE. It's just that it's not a Legendre polynomial because it continues forever (doesn't terminate). Hope that helps!
Still a Legendre function, though, which is kinda "in between" the two subsequent Legendre polynomials, kinda like a fraction is in between two whole numbers ;)
Excellent description Sir. Would it be possible to kindly check and confirm the expression for a4 (i.e a sub 4) at 6.41 minutes?
Nice explanation , thank you sir
Someone tell me where the x comes from in the formulas for y sub odd and y sub even after you do the recusion formula to find the constants. And how is P(x) the same thing as x?
That's just the series solution: the coefficient a_1 is attached to x, the coefficient a_2 is attached to x^2 and so on. And the P(x) you're referring to is actually P_1(x) (i.e. the first Legendre polynomial, which basically means the Legendre polynomial which has 1 as the highest power on x. Similarly, P_2(x) is the 2nd Legendre polynomial, so the Legendre polynomial with 2 as the highest power on x).
This is enlightening. Awesome explanation. This is what iv'e been stuck on for a while in quantum and EnM, because the last math class that's required in our physics major is calc 4 and maybe vector calc, haha so it just pops up out of no where a lot and it's hard picking up on some of these little things. Thanks so much for this, really really saved me a lot of scrambling and frustration.
No problem, glad you like it!
by the way could you please do a video on Bessel equations and functions please thank you
Sure! No problem.
thanks a lot u r the best
Nice compressed video!
can you explane that when we derivative of the series then why not we change the limit of the summation
y' should be from n=1 to infinity and y'' from n=2 to infinity, right? Correct me on this if i'm wrong.
Watching the video all the way from México. Thanks a lot for the great help, keep on with the greath Job. I have a doub't. What happen when I take k=n?
Thank you Gerardo! When k = n, the coefficient becomes zero, and the series terminates at that coefficient (giving you a polynomial solution). So for example, when you have k = 3, the a5 term becomes zero because of the (3-k) factor. I believe I discuss this in the video. Hope that helps!
A silly question but what's the difference between creating a dummy variable and shifting the index of the series? Are these just two methods that do the same thing?
Why doesn't the indexing change when you differentiate the power series?
Because the n = 0 term in the derivative (and the n = 0, 1 terms in the second derivative) is zero anyway. It's not necessary to change the index; I didn't do it so I could keep things simple and consistent.
If m=-2,why didn’t you change the starting point of the first term to -2 too.
I subscribed only because of the never seen before of asking to do it.
P.S- Video is also great.
hi.. thanks fr the video... I have a question.. The series we get After putting the value of y and its derivatives in the differential eqn. Is it necessary to make power of each sum consistent.. What if in some other case the values at m=-1 and m= - 2 doesn't evaluate to zero..
No problem! Yes, the power of each sum has to be consistent. Otherwise, I wouldn't be able to combine the terms like I did in 4:24.
ok, and thanks for responding
how can i calculate without dividing k into odd and even? in the video, you grouped k into 1, 3, 5... and 2, 4, 6,....
But why is this D.E. so special? Other than the terminating series, it seems pretty normal. Why does it have a separate name for it?
Hey, In some books, they have solved legendre equation using frobenius method. Is that right?
Yes, you can solve Legendre's equation with the Frobenius method as well. I find the method used in the video to be easier though.
Faculty of Khan thanks.
It can be solved by the (more general) Frobenius's method as well, but it's like using a sledgehammer to crack a nut ;)
My question is this: How can we use a polynomial from one part of the solution (which converges) and disregard the other part that diverges? Should the two parts of the series be seen as a linear combination of solutions, and we just take the part of the solution that successfully solves the equation for the given value of l?
For anyone else who had the same question (I did), thought about this and came to this conclusion…
Since y is a linear combo of odd and even, if the odd terminates but the even diverges, then a0 must be =0 in order for the series to converge and describe y. If the even terminates, then the odd won’t and a1 must be =0 to converge to y.
can this equation be solved through the method of frobenious
You probably could, but when you look at the indicial equation, your solution would just be r = 0, which is the same as the regular series solution method.
doesnt the initial value of n increase in the differentials.
like n=1 for first derivative and n=2 for the second derivative
It doesn't matter, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2 like you said, but I wanted to keep things consistent.
What book do you use?
Advanced Engineering Math by Kreyszig (did I spell it right) for this series.
Are there any other special sets of polynomials that arise from this differential equation? And yes, I want to see you derive Rodrigues' formula :P
Also, I know Legendre polynomials are orthogonal under the inner product that is the integral of the product from -1 to 1. Is there an easy way to tell this from the differential equation?
at min 2:45 how was it ((1-x)^2 and 2x) expanded?
@Aryaman Prajin Oh haha how couldn't I see that, thanks!
I think....x=0 is an ordinary point...not regular singular point...x=1orx=-1 is regular singular points...
May be I'm wrong....please don't mind..
Awesome video. Thanks a lot!
explained very well
Thank you! Glad you liked it!
If one series terminate and another doesn't, then we still have infinte series representation since y is a combination of odd and even series, so the solution is still unsolved! What's wrong?
It's not unsolved; it's just that one of the solutions is a finite polynomial while the other is an infinite series. The finite polynomial is dubbed the Legendre polynomial.
The solutions for k look similar to formula of polygonal numbers
We study this in my junior year for chemical engineers
how do you put values of a0=0.5 and a1=-3/2
I just used them so I could get expressions for the Legendre polynomials. For those exact values of a0, we have the Legendre polynomials. Hope that helps!
@@FacultyofKhan if we use different value of a0 or a1.. that will derived different answers. it will not be equal anymore using rodriguez formula. please enlighten us
There's a slight misleading information here. There is a normalization constant acting upon the solution of the 'y1' and 'y2'. You also have to mention them. The arbitrary constant doesn't make up the value.
I don't remember learning any of these in my class lol
omg thank you so much you are the best
I just like ur speed.
Thanks a bunch!🤗🤗🤗
how i do about X0=1?
At 8:28, why is y even = P2(x)?
P2(x) corresponds to k = 2, and y_even represents the terms that come from using even indices k on the recursion relation. Also, for a0 = -1/2, y_even becomes the second-order Legendre polynomial, which I've denoted as P2. Hope that helps!
Eminem of mathematics .
the summation of series is zero ...so how can you justify it by saying that all the coefficients that are associated with the increasing power of the variable x is zero .... can it not be the case that terms in the series have a distribution of positive and negative values ?
I did subscribed cause you are special too :D
where 900pp?
lul
what if k=6?
legendre equation is used to obtain the potential due to gravitational system where there are no charges and masses . obtain the series solution to get potential
I believe so! I remember it was used in my Electricity & Magnetism course.
give me name of the video
or else a written solution
Well, I haven't made a video on that right now. But you can look here starting at section 2.2:
www.physics.usu.edu/Wheeler/EM3600/Notes08SolvingForPotentials.pdf
Hope that helps!
so,who's the real khan?
There is a mistake in this video if the summation becomes zero till m=2 then the summation should start from m=2
It doesn't matter if the summation is zero for m = 0 and m = 1; I could start it at m = 2 like you said but it doesn't make a difference since the m = 0 and m = 1 terms are zero anyway.
Thanks a lot!
I have this doubt that when we shift the index of the first series to make x^s common then we do not shift the index of the rest of the equation. When we differentiate the series twice we are supposed to get A1 constant for first differentiation and A2 term for second. Then why haven't you showed it. Just for the sake of getting X^s common we cannot change the series according to our convenience.
I'm not sure I fully understand your question, but I'll try to explain it according to what I pieced together. I feel like you've asked 2 questions:
1. When you shift the index of the 1st series (around 3:00), why don't you shift the index of all the other series? Answer: The reason is that for now, I just want to make sure the x's are all raised to the same power, so it's easier to change the first series.
2. Why don't you change the starting value of n when you differentiate the series solution y = sum from n = 0 of an*x^n? Answer: I've discussed it in previous responses, but it doesn't matter, since the term corresponding to n=0 for the first derivative and those corresponding to n=0,1 for the second derivative are all zero. I could have started them at n=1 and n=2, but I wanted to keep things consistent.
can you now make a video on how to get girls to like single and lonely mechanical engineers ? thanks
step 1. start lifting, step 2. wait 1-3 years, step 3. learn guitar/piano , step 4. buy a puppy, step 5 walk. step 6. talk, step 7, get her phone nubmer , step 8 sleep, and repeat
so, mechanical engineers are lonely in all parts of the world?
When tf did this turn into a pickup channel?
Or start picking up lonely mechanical engineer girls ;) You can always impress them with solving some crazy ODE :)
I wouldn't go with muscles myself, because chick that love muscles are usually empty and boring as hell :P so you don't want to hang out with them anyway, trust me :q
@@3631162 Relax and lighten up cringehere. So just because textbooks show photos of Laplace in formal coat and not laughing does not mean he did not have fun. Think about that.
your voice is like rhydum..you make me asleep....its my test tomorrow you r responsible for my dirty gradez
lol well I can't really change my voice now; still, I hope you at least found the lesson useful. You can also turn on captions and put it on mute if the voice is too boring.
yeah thats better idea....thanks
i really cant understand the last part.. can someone please explain to me in a simple way
Sure! If you go back to 8:06, you can see the recursion relation at the top (a_n+2 as a function of a_n).
All the last part is saying is that when k is an ODD number, the coefficients a_n for ODD values of n continue only up till a certain point, because when n reaches k, you can see that the numerator of the recursion relation immediately becomes zero. Now if one of the coefficients (a_n or a_k) reaches zero, all coefficients that come after that (e.g. a_n+2, a_n+4, etc) for odd values of n will also be zero. This means that the 'odd' index series would terminate, and we end up with a polynomial involving odd powers of x (e.g. P_1, P_3 etc).
The same logic applies to EVEN values of k, in which the coefficients a_n for EVEN values of n eventually terminate.
If you have any more questions, please ask!
+Professor Khan Thank you Prof!! I really appreciate your respone.. Thank you soo muuchh!!😃
No worries! Glad I could help!
What is an ODE?!
4:55
magnificent!
great vidoes!!!!!!!!!
Thank you very much!
❤
do you have any relation to Khan academy or your name itself is Khan?
謝謝！
why a subcription 0 is taken as half?
I don't follow your question. What do you mean by subscription? Can you point to a place in the video where I talk about this?
10:17 Well it worked.
goodnight all,, -,-
^
so fast explanation, hard to swallow T_T
ur too good ...but robotic voice