How can I find two linearly independent solutions for 4xy''+2y'+y=0?

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    Frobenius Series
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Discussion Overview

The discussion revolves around finding two linearly independent solutions for the differential equation \(4xy'' + 2y' + y = 0\) using the Frobenius method. Participants explore various approaches to solving the equation, including series solutions and transformations, while addressing the challenges faced by the original poster.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • The original poster expresses difficulty in solving the differential equation and requests assistance.
  • Some participants inquire about the methods the original poster has attempted, emphasizing the need for clarity on their progress.
  • One participant suggests starting with a power series solution and finding the derivatives \(y'\) and \(y''\) to substitute back into the equation.
  • Another participant mentions that dividing by \(4x\) is unnecessary and reiterates the importance of assuming a solution to find the derivatives.
  • A later reply introduces an alternative approach by substituting \(x=t^2\) to simplify the differential equation, noting that this leads to a closed-form solution that can be compared to the series solution.
  • Concerns are raised about the original poster's understanding of calculus, particularly regarding the differentiation of power series.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the differential equation. Multiple competing views and methods are presented, with some participants advocating for the Frobenius method while others suggest alternative transformations.

Contextual Notes

There are indications of missing foundational knowledge in calculus from the original poster, which may limit their ability to engage with the problem effectively. The discussion also reflects varying levels of familiarity with the Frobenius method and differential equations among participants.

Who May Find This Useful

This discussion may be useful for individuals learning about differential equations, particularly those interested in series solutions and the Frobenius method, as well as those seeking alternative methods for solving such equations.

rapwaydown
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the question is
" find two linearly independent of frebenius series solutions for 4xy''+2y'+y=0"

I try almost everything to slove this, but could't figure it
any help is appercaited

thank you
 
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Well, what have you done on it? You say you've tried "almost everything". Okay, what have you tried?
 
HallsofIvy said:
Well, what have you done on it? You say you've tried "almost everything". Okay, what have you tried?


well, i divided the whole thing by 4x, trying to get it to the general format, but it didnt work out.
 
In other words, you really haven't done anything!

Start by writing
y= \sum_{n=0}^\infty a_n x^{n+c}
Find y' and y" from that and put them into the equation. What do you get?
 
HallsofIvy said:
In other words, you really haven't done anything!

Start by writing
y= \sum_{n=0}^\infty a_n x^{n+c}
Find y' and y" from that and put them into the equation. What do you get?

i don't know how to find y' and y''
im slef learning def.Q
thanks for the reply tho.
 
HallsofIvy said:
In other words, you really haven't done anything!

Start by writing
y= \sum_{n=0}^\infty a_n x^{n+c}
Find y' and y" from that and put them into the equation. What do you get?

by the way, i don't think y' and y'' is needed here
i divided the whole thing by 4x, then slove for the indicial,
which are r=0,-.5
but i don't know what to do from this point on,
i think i need to find the equation for cn, but don't know how.
can you help
 
You have to assume a solution as you have been advised
You don't need to divide by 4x.
When you assume the solution you will find d second and first derivatives of y, you will then subtitute into the equation.
I solved it and that's the way it goes.
 
Though not the question it is interesting to know that this DE has a solution in closed form. One can find this by substituting x=t^2 in the equation. A very simple DE will appear and can be solved directly. Doing the inverse substituting on this solution gives the result of the original DE. This solution can then be compared to the series solution.
 
rapwaydown said:
i don't know how to find y' and y''
im slef learning def.Q
thanks for the reply tho.
It's probably not a good idea to try to learn differential equations, by your self or not, if you do not know Calculus!

Are you seriously saying that you do not know how to find the derivatives of xn?

And, to even attempt a problem like this you should have had enough Calculus to know that a power series is "term by term" differentiable inside its radius of convergence.
 

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