What is Frobenius: Definition and 105 Discussions

Frobenius is a surname. Notable people with the surname include:

Ferdinand Georg Frobenius (1849–1917), mathematician
Frobenius algebra
Frobenius endomorphism
Frobenius inner product
Frobenius norm
Frobenius method
Frobenius group
Frobenius theorem (differential topology)
Georg Ludwig Frobenius (1566–1645), German publisher
Johannes Frobenius (1460–1527), publisher and printer in Basel
Hieronymus Frobenius (1501–1563), publisher and printer in Basel, son of Johannes
Ambrosius Frobenius (1537–1602), publisher and printer in Basel, son of Hieronymus
Leo Frobenius (1873–1938), ethnographer
Nikolaj Frobenius (born 1965), Norwegian writer and screenwriter
August Sigmund Frobenius (died 1741), German chemist

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  1. S

    Use the method of Frobenius to

    Homework Statement Use the method of Frobenius to find one solution near x = 0 of x^2 y'' + (x^2 + 2x)y' - 2y = 0. Homework Equations Summation notation. (The solution in my book is using the "dot dot dot" notation which I personally hate a lot since it requires writing a finite...
  2. B

    Legendre's Diff Eq using Frobenius

    Homework Statement Show, by direct examination of the Frobenius series solution to Legendre's differential equation that; P_n(x) = \sum_{k=0}^{N} \frac{(-1)^k(2n-k)!} {2^n k! (n-k)! (n-2k)!}x^{n-2k} ;\ N=\frac{n}{2}\ \mathrm{n\ even,}\ N=\frac{n-1}{2}\ \mathrm{n\ odd} Write down the first...
  3. J

    Differential Equations, Frobenius' Method

    Homework Statement Find the indicial roots of the following Differential Equation: xy'' - y' + x3y = 0 Homework Equations y = Ʃ[n=0 to infinity]cnxn+r y' = Ʃ[n=0 to infinity](n+r)cnxn+r-1 y'' = Ʃ[n=0 to infinity](r+r)(n+r-1)cnxn+r-2 The Attempt at a Solution Plugging these values into the...
  4. C

    Finding a solution to this equation using Frobenius method

    Hi, I have this equation to solve. y'' + (1/x)y' + [(x^2) + k + (m^2 / x^2)]y = 0 now, I've tried to solve this using frobenius method but cannot formulate a solution. I have that a_(n+4) = [-ka_(n+2) - a_(n)] / [n^2 +/- 2inm] is my recurrence relation, but now I'm stuck and...
  5. O

    Using the Frobenius Method - 2nd order DE

    Using the Frobenius Method -- 2nd order DE Homework Statement y"+(1/sinx)y'+((1-x)/x^2)y=0 Find the indicial equation and forms of two linearly independent expansions about x=0. Don't find the coefficents. Homework Equations The Attempt at a Solution The singular points at...
  6. T

    Solve Frobenius Series Homework for 2 Independent Solutions

    Homework Statement The origin is a regular singular point of the equation 2x^2 y'' + xy' - (x+1)y =0. Find 2 independent solutions which are Frobenius series in x. The Attempt at a Solution Substituting y = \sum_{n=0}^{\infty} a_n x^{n + \sigma} eventually gives (2\sigma(\sigma - 1)...
  7. Telemachus

    Solve Frobenius Method Homework: x(1-x)y''+2(1-2x)y'-2y=0

    Homework Statement Hi. Well, I need some help with this problem. I have to solve x(1-x)y''+2(1-2x)y'-2y=0 (1) Using Frobenius method around zero. So proposing y=\Sigma_{n=0}^\infty a_n x^{n+\alpha}, differentiating and replacing in (1): x(1-x)y''+2(1-2x)y'-2y= =\Sigma_{n=0}^\infty...
  8. R

    Differential Equation, Frobenius method

    Homework Statement Find two linearly independent power series solutions for xy" - y' + xy = 0 using the Frobenius method. Homework Equations The Attempt at a Solution solving for the indicial roots I got: -> r(r-2) = 0 r = 0, 2 for the recursion formula I got...
  9. O

    Extended version of associated Legendre eq. Can I use the Frobenius method?

    Hello, Let us assume we have a differential equation ( \frac{d^2}{dx^2}+ \frac{2}{1+1/n}\Theta_n^{n-1} - \nu^2 ) y = 0 where \Theta_n = (1-\mu^2)^{1/(n+1)} is a function of \mu which is a function of x : \frac{d\mu}{dx} = \Theta_n^n = (1-\mu^2)^{n/(n+1)} In terms of \mu...
  10. N

    Confused about the theory behind Frobenius' Method

    Hello, My course is a bit vague on this topic. 1 First of all it states: if the differential equation A(x)y''(x) + B(x)y'(x) + C(x)y(x) = 0 with A, B, C analytical has A(a) = 0 but B(a) and C(a) NOT both zero, then a is a singular point and we can't solve it with a power series. Now: what...
  11. P

    Method of Frobenius with exponential coefficients

    I know how to do Frobenius on variable coefficient ODE's but only when the coefficients are powers of the independent variable. Can I do method of Frobenius on something like: y'' + e-xy = 0 ? What form would I assume a solution of? Just the regular y=sum(Akxk+r ? Thanks for the help!
  12. J

    Frobenius power series repeated roots

    Could someone please explain the y2 solution for repeated roots in Frobenius method where y2=y1lnx+xs \Sigma CnxnI am struggling to figure out how to solve this
  13. Saladsamurai

    Diff EQs: Power Series vs Frobenius Solutions: Difference?

    This is a pretty general conceptual question. I was just doing some reviewing for a test, and it occurred to me that if I were not told specifically to use Frobenius method on an equation, I might try to Power series solve it and vice versa. Can we talk about the difference a bit? We apply...
  14. J

    Power series when to use Frobenius method

    Hi, I'm new to the forum and need some help regarding my calc class. Any help you could provide would be greatly appreciated. In doing a power series series solution when should I use the frobenius method and when should I use the simple power series method. The simple method seems a little...
  15. Saladsamurai

    Frobenius Equation 1: Almost there

    Homework Statement Hi :smile: I think I am making some good progress on this one, but I am unsure of what the next step is? Can someone give a nudge in the right direction?
  16. Saladsamurai

    Questions about Frobenius Method

    Homework Statement A Frobenius equation takes the form y'' + p(x)y' +q(x)y \qquad (1) but for the sake of definiteness, let's take a particular example that my text uses: 6x^2y'' + 7xy' - (1 + x^2)y = 0 \qquad(2) We seek the solution in the form of y(x) = \sum_0^{\infty}x^{n+r}\qquad(3)...
  17. T

    Frobenius method indical roots differing by a postive integer

    Homework Statement For what values of the constant K does the differential equation y"-(1/4 +K/x)y=0 (0<X< infinity) have a nontrival solution vanishing at x=0 and x= infinity ? Homework Equations Hints that were suggested for my prof were: For large x...
  18. TheFerruccio

    What is the next step in solving this Frobenius method problem?

    I am having an issue understanding the Frobenius method. I plan on posting multiple questions in this thread as I go. I searched these forums and saw some other threads pointing to it, but the questions were slightly different. I am able to get through the problem to a certain extent, but...
  19. 0

    Solving a Frobenius Equation: Finding a Regular Point at x = 0

    hey i am stuck on this question for my ode course its using frobunius 4. show that the equation yii + 1/x yi + (1-1/(4*x^2))y = 0 has a regual point at x=0 using the method of frobenius assuming a solution of the form y=\sum ar xc+r show that the idical equation is c^2=1/4...
  20. B

    Frobenius method non-polynomial coeficients.

    Homework Statement Find the first 3 terms of two independent series solution for the DE. xy''+2xy'+(6e^x)y=0 Homework Equations Frobenius method. case r1-r2=integer The Attempt at a Solution I found that 0 is a regular singular point. I found the indicial equation and found the...
  21. Z

    Frobenius method solution for linear ODE

    I've been given the ODE: x^2 u''-x (x u'-u)=0 Solve. It's suppose to be an example in which a logarithmic term is required for the general solution. I would be glad if someone could look at what I've done and see if my solution is correct / incorrect. Thank you in advance for your time...
  22. J

    Solving ODE with Frobenius Form Solution

    I'm finding a solution to and ODE using a Frobenius form solution, I have worked through the question and have ended up with a recurrence of the form; a(k) = -2/(2k-1) a(k-1) I'm trying to find a general reccurence in terms of a(0) but am finding it a bit difficult, I know it has to have a...
  23. N

    Frobenius method aplied to simple harmonic oscillator problem

    Homework Statement P4-1. The Method of Frobenius: Sines and Cosines. The solutions to the differential equation y"+ y = 0 can be expressed in terms of our familiar sine and cosine: y(x) = Acos(x) + Bsin(x) . Use the Method of Frobenius to solve the above differential equation for the even...
  24. Z

    How can the Frobenius method be used to find a solution for a second-order ODE?

    Homework Statement Use the Frobenius method, for an expansion about x=0, to find ONE solution of xy''+y'+(1/4)y=0 Homework Equations The Attempt at a Solution starting with an assumption of y1=\sumanxn+r and plugging it into the ODE, i found y=\frac{-1}{4}\suma0xn/(n!)2...
  25. H

    Solving Frobenius Method Problems: Tips & Tricks

    Am I missing something in the Frobenius method?? Homework Statement Use the method of Frobenius to find solutions near x= 0 of each of the differential equations. The Attempt at a Solution x^2 y'' + (2x^2 + 3x)y' + (x-(5/4))y = 0 My work is as follows: http://imgur.com/3XvP4.jpg...
  26. R

    Easy ODE to solve using frobenius method help?

    I can't seem to figure this one out: y'' + (1/x)*y' -c*y = -c*y0 where c is just a constant. if someone could please go through the steps of the frobenius method, i would really appreciate it! thanks.
  27. I

    Frobenius solution to a diff-eq

    Homework Statement Find 2 Frobenius series solutions to the following differential equation: 2xy'' + 3y' - y = 0 Homework Equations The Attempt at a Solution I got r = -1/2 and 0 as roots. Recurrence relation...
  28. R

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

    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
  29. I

    Frobenius Series: The Significance of Non-Zero First Terms

    Why do we assume that the first term c0 in a frobenius series cannot equal 0? Thanks!
  30. I

    Frobenius Method of Solving ODE

    Homework Statement Find two series solutions of the given DE about the regular point x = 0. 2xy'' + 5y' + xy = 0 Homework Equations The answer is in the form y = SUM(cnxn+r) where SUM is the sum from n = 0 to infinity. The Attempt at a Solution This questions got me stumped. I'm...
  31. G

    Jordan Can. Form of Frobenius map

    Hello all, I am trying to solve this exercise here: Let \phi denote the Frobenius map x |-> x^p on the finite field F_{p^n}. Determine the Jordan canonical form (over a field containing all the eigenvalues) for \phi considered as an F_p-linear transformation of the n-dimensional F_p-vector...
  32. M

    Solving ODEs w/ Frobenius Method: Q on b, c as Funcs of x

    I have been looking at the Frobenius method for solving ODEs of the form. I have a few questions on it. (x^2)y'' + xby' + cy = 0 If b and c are functions of x, does one use the Frobenius method, where as if they are constants, it is an Euler Cauchy equation and you use y = x^r ...
  33. M

    Frobenius Method again

    Assume that x=0 is a regular singular for x2y" + xp(x)y' + q(x)y = 0 and the indicial equation has equal roots \lambda = \lambda_1 = \lambda_2 The first solution is alway known to be of the form y_1(x) = x^{\lambda_1}\sum a_n x^n Although tedious, I know how to obtain the second...
  34. M

    A harder Frobenius Differential

    Homework Statement xu'' + 2y' + xy = 0 Homework Equations http://en.wikipedia.org/wiki/Frobenius_method The Attempt at a Solution Ok, so i have managed the Frobenius method in the past, but this seems harder... \sum_{n=0}^{\infty} a_n(n+c)(n+c-1)x^{\ n+c-2}...
  35. D

    Quick Frobenius Question

    Hello, I was just wondering, if I have a differential equation that has two regular singular points, and I am asked for the general solution, do I need to use the Frobenius method about each point seperately? I suspect that I do I just want to clarify. Thanks
  36. B

    Finding Series Solutions Using Method of Frobenius

    Homework Statement Using method of Frobenius, find a series solution to the following differential equation: x^2\frac{d^2y(x)}{dx^2} + 4x\frac{dy(x)}{dx} + xy(x) = 0 Homework Equations y(x) = \sum_{n = 0}^\infty C_{n} x^{n + s} The Attempt at a Solution y(x) =...
  37. B

    Understanding Frobenius Method & ODEs: When to Use Set 1 or Set 2

    I'm a little confused with ODEs. After two weeks of trying to figure out Frobenius I have finally realized that there seems to be two different power set used by all of my three books for the y substitution but I am unsure when to use either one. Here are the two sets that I'm talking about...
  38. V

    Understanding Ring Relationships and the Frobenius Automorphism

    I'm having trouble with what seemed like a trivial problem at first, but now I'm rather stuck. If R is a ring with xy=-yx for any x,y from the ring, xyz+xyz=0 must be true for any x,y,z from the ring. I'm trying to show why that is. Letting y=x yields x^2+x^2=0. Thus then breaking it up into...
  39. G

    Solving 3xy" - 4y' - xy = 0 with Frobenius

    I'm really getting stuck at this and I'm trying to read on it but it's confusing. I need just a start-up for this equation which is to be solved with the method of frobenius. 3xy" - 4y' - xy = 0 Just need a start. Any help is appreciated. Thank you
  40. D

    Solving Differential Equation with Fröbenius Method

    I'm trying to solve the differential equation: x^2y''+2xy'+(x^2-2)y=0 using the Fröbenius method. So I want a solution on the form y=\sum_{n=0}^\infty a_{n}x^{n+s} After finding derivatives of y, inserting into my ODE, and after some rearranging: \sum_{n=0}^\infty...
  41. G

    I hate Frobenius series, can anyone help

    Please find attached the problem I am having difficulty with its part (b) i need help with. Cheers, Dave:cry:
  42. S

    Legendre's equation with Frobenius method

    I know you don't need Frobies method per se, but he wanted us to practice, well I got the right recursion formula, but I realize I got to it wrong...After I plugged in the assumed series solution and all its derivatives and stuff and got that big long equation, it had two terms out in front and...
  43. J

    Differential equation with frobenius method

    Im trying to solve y''(x) + xy'(x) + y = 0 For the first solution i get with frobenius method y_1(x) = sum_{p=0}^\infty \frac{a_0(-1)^px^(2p+1)}{(2p+1)!} Im not sure if its correct but i think so. This must be the taylor series for some function. Can someone help me with...
  44. F

    How to solve a second order linear homeogeneous ODE with Frobenius?

    A simple question i think although i can't find in any books What do u when u are solving a second order linear hoemoeneous differential equation with frobenius and there is no shift. (X^2)(y^{''}) (-6y)=0 it should be normal minus -6y I only know what to do if there...
  45. A

    Frobenius Method: Finding Smaller Root with Larger

    for the forbenius method, if the roots to the indical equation differ by an integer, why do you always have to take the larger root to find the smaller root?
  46. A

    Is the legendre equation an example of a frobenius equation?

    is the legendre equation an example of a frobenius equation?
  47. A

    Why is A0=1 in the Frobenius Method?

    for the frobenius method, why do you suppose that a0=1?
  48. T

    Frobenius' Theorem: Characterization & Proof Difficulty

    I need to understand a certain characterization of Frobenius' Theorem, part of which contains the following statement: \nabla_{[a}\xi_{b]}=\xi_{[a}v_{b]} for some dual vector field v_{b} if and only if \xi_{[a}\nabla_{b}\xi_{c]}=0, where \xi^a\xi_a\neq 0. Is it obvious, or difficult to prove...
  49. E

    Solving ODEs with Frobenius Method: Troubleshooting and Tips

    i need your help...i had to solve some ode using the frobenius method but half-way through i stuck... for the first ode i didn't know what to do when i had to multiply the sinx and cosx with y,y',y" ii)i had a problem because of the cosx and i couldn't equate the series...--->second ode...
  50. J

    Differential Equation: Frobenius

    My problem: find the first solution and use it to find the second solution for x^2*y"-x*y'+(x^2+1)y=0 assuming y=summation from n=0 to infinity for An*x^n+r substituting and solving gives me r=1 and a general equation: An=A(n-2)/((n+r)*(n+r-2)+1) for n >= 2 plugging r into my...
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