ODE:: y'' - xy' + 2y=0 :: Power Series Solution about an Ordinary Point
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- เผยแพร่เมื่อ 21 ก.ย. 2024
- Here, we derive two linearly independent solutions of a differential equation y''-xy'+2y=0 using a power series expansion about an ordinary point.
An ordinary point is one that is not singular. A singular point is one that make the coefficient of y'' zero. As there are no singular points in this example we see every point is ordinary.
This is the standard approach to finding solutions about ordinary points.
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My name is Jonathan, and I have taught thousands of students. Solving a differential equation is exciting to me every single time. The thrill of the problem is unique for each one, and I hope to provide useful strategies and techniques for using power series to solve differential equations for you to implement.
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@@JonathanWaltersDrDub What happens if there is initial condition given? y(0)=1, y'(0)=2 Approximate the value of y(2) using power series solution. Approximate the value of y using only up to 8th degree term of the series. Write your answers in 4 decimal places.
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Glad to help! Do well!
@@JonathanWaltersDrDub thank you!🙏
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Me: 'On verge of absolute breakdown'
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Me: "Breakdown averted'
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Thank you!!! Can you help with the general recurrence relationship for this problem and the general idea(general recurrence relationship )?
13:15 - shouldnt we have (2ak) before the last sum?
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if we want to find the solution of the differential equation y''-3xy'=0 with the help of the power series around x_0=1, then the reccurrence relation to obtained is?
Simple and clear.....thank you
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at the end shouldn't the third term c_0x^2 not have been included since the series started at k=1 and not k=0?
this was super helpful, thanks a lot!
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Simplified and helpful💗
I'm glad this helped!
Question please:
Why do we have K greater or equal than one in the recurrence relation? Check on the 16th to 17th minute
Sorry that I missed this! k is greater or equal to 1 because the series terms start at k=1. Since the recurrence is from the terms inside the series it's only valid for the k values for which the series is defined.
imagina no enterder nada en lo absoluto de estas series y de remate no entiendes ingles :( jajajaja gracias me ayudara para mi clase(espero)
¡Me imagino que es muy difícil! ¡Mantener el trabajo duro!
Can we reindex but have the same variable we had originally. For example, could we still use n as long as we are reindexing instead of k?
That is okay but you just have to be very careful. It would be like doing a u-substitution in an integral but still just using x.
simply??? hm.... (thx for the awesome video)
Thanks for your support!
At 15:00, does the answer have to be linearly independent? Or are we just assuming it is so that we can solve it?
At 15:00 we are using the fact that 1, x, x^2, x^3, x^4, ... etc... are linearly independent to set the coefficients = 0 so we can find the recurrence relation.
Thank u very much sir.
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Do you do the solution in this process? y''+xy'-3y=4,2x y(0)=0 y(1)=1,9 Δt=0,25 dissolve on paper please at least
Hello!
𝒚′′ + 𝒙^2𝒚′ + 𝒙𝒚 = 𝟎 y(0)=0, 𝑦′(0) = 1
Solve the differential equation with the power series. Can you solve this question?
did you solve it?
Hold on what happened to the X he forgot the X
I think I almost threw up when I saw your comment. I got a little panicked to say the least
! 😅. Maybe check 5:18?
Which x
Thank you. You made me understand power series
I’m happy to hear it’s clicking for you now!