Frobenius Definition and 107 Threads

Frobenius's theorem gives necessary and sufficient conditions for smooth distributions ##\mathcal D## defined on a ##n##-dimensional smooth manifold to be completely integrable. Now consider a smooth frame field given by ##n## linearly independent smooth vector fields. I suppose Frobenius's...

Hi, starting from this thread, I'd like to clarify some mathematical aspects related to the notion of hypersurface orthogonality condition for a congruence. Let's start from a congruence filling the entire manifold (e.g. spacetime). The condition to be hypersurface orthogonal basically means...

Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1-form ##\omega##, namely that ##d\omega = \omega \wedge \alpha## is actually equivalent to the existence of smooth maps ##f## and ##g## such that ##\omega = fdg##. I found this About...

I have no problems with solving this exercise, but my solution disagrees slightly with that given in the answers in the back of the book, and I do not know who's correct. First, we rewrite the equation as $$x''+\frac{3}{2t}x'-\frac{(1+t)}{2t^2}x=0.\tag1$$ We recognize that this is so-called...

Hi, reading this old thread I'd like a clarification about the following: Fermi Normal hypersurface at an event on a comoving FLRW worldline is defined by the collection of spacetime orthogonal geodesics. Such geodesics should be spacelike since they are orthogonal to the timelike comoving...

I'm reading a book called Asymptotic Methods and Perturbation Theory, and I came across a derivation that I just couldn't follow. Maybe its simple and I am missing something. Equation 3.3.3b below. y(x) takes the form A(x)*(x-x0)^α and A(x) is expanded in a taylor series.

Apologies for potentially being imprecise and clunky, but I'm trying understand integrability of the following Hamiltonian $$H(x,p)=\langle p,f(x) \rangle$$ on a 2n dimensional vector space $$T^{\ast}\mathcal{M} =\mathbb{R}^{2n}.$$ Clearly this is just the 1-form $$\theta_{(x,p)} =...

I have to find 2 solutions of this Bessel's function using a power series. ##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0## I'm using Frobenius method. What I did so far I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##...

Hi, searching on PF I found this old post Global simultaneity surfaces. I read the book "General Relativity for Mathematicians"- Sachs and Wu section 2.3 - Reference frames (see the page attached). They define a congruence of worldlines as 'proper time synchronizable' iff there exist a...

Hi, reading the Landau book 'The Classical theory of Field - vol 2' a doubt arised to me about the definition of synchronous reference system (a.k.a. synchronous coordinate chart). Consider a generic spacetime endowed with a metric ##g_{ab}## and take the (unique) covariant derivative operator...

I'm having a hard time grasping the concept of reducing the two recursive relations at the end of the frobenius method. For example, 2xy''+y'+y=0 after going through all the math i get y1(x) = C1[1-x+1/6*x^2-1/90*x^3+...] y2(x) = C2x^1/2[1-1/3*x+1/30*x^2-1/630*x^3+...] I know those are right...

Homework Statement i have been trying to learn bessel function for some time now but to not much help firstly, i don't even understand why frobenius method works why does adding a factor of x^r help to fix the singularity problem. i saw answers on google like as not all function can be...

I have calculated that a matrix has a Frobenius norm of 1.45, however I cannot find any text on the web that states whether this is an ill-posed or well-posed indication. Is there a rule for Frobenius norms that directly relates to well- and ill-posed matrices? Thanks

Homework Statement Use Forbenius' method to solve this DE: $$ 5x^2y''+xy'+(x^3-1)y=0$$ Homework Equations Seek power series solution in the form ##y=\sum _{n=0}^{\infty } a_n x^{n+r}##, ##a_0\neq0## The Attempt at a Solution Sub in the ansatz y, get $$ \sum _{n=0}^{\infty...

Homework Statement If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations included in attempt The Attempt at a Solution I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though. Whenever I expand for m=0 for the a0 I end...

Hi I am supposed to find solution of $$xy''+y'+xy=0$$ but i am left with reversing this equation. i am studying solution of a differential equation by series now and I cannot reverse a series in the form of: $$ J(x)=1-\frac{1}{x^2} +\frac{3x^4}{32} - \frac{5x^6}{576} ...$$ $$...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of the proof of Theorem 1.4 ... ... Theorem 1.4 reads as follows: Questions 1(a) and 1(b) In...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of the proof of Theorem 1.4 ... ... Theorem 1.4 reads as follows: Questions 1(a) and 1(b) In the above...

Frobenius method - recurrance relation question If, using the Frobenius method, I get a 3 term recurrence relation of the form $a_{j+2} = a_j .f(k,j) + a_{j-2}. g(k,j)$ ( j even), how do I treat the $a_{j-2}$ term at first? I have found $a_1 = 0$, but how do I find a value for $a_{-2}$ so as...

So, when you use the Frobenius method on a differential equation, you assume a solution Σa_k*x^(k+s). Sometimes you get more than one solution for s in the indicial equation. Is the sum of these two solutions you get from evaluating the rest of the problem with each s solution the...

Homework Statement Solve \begin{equation*} 36x^2y''+(5-9x^2)y=0 \end{equation*} using the Frobenius method Homework Equations Assume a solution of the form \begin{equation*} y=\sum_{n=0}^{\infty}{a_nx^{n+s}} \end{equation*} then \begin{equation*}...

Homework Statement Given the differential equation (\sin x)y'' + xy' + (x - \frac{1}{2})y = 0 a) Determine all the regular singular points of the equation b) Determine the indicial equation corresponding to each regular point c) Determine the form of the two linearly independent solutions...

Homework Statement Hello all, I have a quick question, I'm solving a d.e using the Frobenius method and I have the indicial equation: C1(2r-1)(r-1)+C2x(r)(2r+1)=0 Where c1 and c2 are arbitrary constants not equal to zero. Homework EquationsThe Attempt at a Solution My question is, what are...

After determining that x = 0 is a regular singular point of this equation, the frobenius method allows you to assume that y = Σanxn + r. Then I can take the first and second derivative of this assumption and plug it into the DE and begin solving with the general method: Multiply the...

Homework Statement I'm asked to solve the Hermite Differential Equation y''(x) - 2 x y'(x) + \lambda y(x) = 0 using the Frobenius method 2. Homework Equations I am to assume the solution is in the form y(x) = \sum a_n x^{n+r} where r are the roots of the indicial equation that in this...

Hi, I'm having trouble with this one. Homework Statement Find a particular solution of the second-order homogeneous lineal differential equation x^2y'' + xy' - y = 0 taking in account that x = 0 is a regular singular point and performing a power series expansion. Homework...

∑Homework Statement Use the method of Frobenius, constructing a power series about x = 0, to find the general solution of this equation (retain terms up to and including those in square brackets): 4xy'' + 2y' + y = 0 [x7/2] Note: the solution can be written in closed form, can you...

the actual problem is to show that the given matrix is similar to companion matrix here is the companion matrix Companion matrix - Wikipedia, the free encyclopedia ---------------- i know that if same frobenius canonical form then similar but i don't even know how to find the frobenius...

By using frobenius method I find the roots of the indicial equation of a 4th order ODE to be 0, 1, 1, 2 Now, what is the form of the corresponding series solution of this equation with log terms?

Homework Statement Use method of Frobenius to solve this equation: ##y''(x)-y'(x)=x## Homework Equations ------ The Attempt at a Solution Seek an answer of the form ##y=\sum _{n=0}^{\infty } a_n x^{n+r}## Plug into the equation to get... ##\sum _{n=0}^{\infty } a_{n+1} (n+r)...

Homework Statement I want to find two linearly independent solutions of $$ x^{2}y''-2x^{2}y'+(4x-2)y=0. $$ The Attempt at a Solution The roots to the indicial polynomial are ##r_{1}=2## and ##r_{2}=-1##. I found one solution which was ##x^{2}## and I am having trouble finding the...

Homework Statement Consider x^2y''-xy'+n^2y=0 where n is a constant. a) find two linearly independent solutions in the form of a Frobenius series, initially keeping at least the first 3 terms. Can you find the solution to all orders? b) for n=1 you shouild find only one linearly...

Homework Statement Solve x(1-x)\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0 using the Frobenius Method. Homework Equations R(x)\frac{d^{2}y}{dx^{2}}+\frac{1}{x}P(x)\frac{dy}{dx}+\frac{1}{x^{2}}V(x)y=0 R_{0}s(s-1)+P_{0}s+V_{0}=0 y=\sum^{∞}_{m=0}a_{m}x^{m+s}...

Not sure how this can be done. can anyone help?

Homework Statement Prove ∥A∥F =√trace(ATA), for all A ∈ R m×n Where T= transpose Homework Equations The Attempt at a Solution I tried and i just can prove it by using numerical method. Is there anyway to prove the equation in a correct way?

Homework Statement The task is to find an analytic solution to the O.D.E 4(1-x^2)y''-y=0 \hspace{20mm} y'(1)=1 by using an appropriate series solution about x=1. The Attempt at a Solution The singularity at x=1 is regular, which makes me think the Frobenius method is what's meant by...

Hi guys. Most of my texts have the standard proof of Frobenius' theorem (both the vector field and differential forms versions) and through multiple indirect equivalences conclude that ##\omega \wedge d\omega = 0## implies (locally) that ##\omega = \alpha d\beta## where ##\omega## is a 1-form...

Homework Statement what is the limit of (4x^2-1)/(4x^2) when x→0 Homework Equations In order to find the Indicial Equation, do I need to take the limit of p(x) and q(x), the non-constant coefficients? If so, can the limit of this function be found using LH Rule? The Attempt at a...

Hello, I am trying to write a mtlab code to compute Frobenius norm of an mxn matrix A. defined by ||A||_{F} = \sqrt{ \sum_{i=1}^m \sum_{j=1}^n a^{2}_{i,j}} I have so far written this code, but it does not work, if anyone can help /guide me to the right path, would be greatly...

Homework Statement The function satisfies the differential equation f''(x) = xf(x) and has boundary conditions f(0) = 1 and f'(0) = 1 Use Frobenius method to solve for f(x) with a taylor expansion of f(x) up to the quartic term a4x4 Homework Equations f(x) = a0 + a1x + a2x2 + a3x3 + a4x4...

Ok here's a funny ODE to solve: xy'' + (1-2x)y' + (x-1)y = 0 clearly a straight forward power series substitution won't work here since we have a regular singularity at x = 0 so try the frobenius method by expanding around x = 0. Assume y = \sum_{m=0}^{\infty} a_mx^{m+r} is a solution where...

Solve xy'= y using frobenius method The explanation given in the book is very confusing can somebody explain in simple method. Thanks

Homework Statement y''+4xy'+(4x^2+2)y=0 find the basis of solutions using the frobenius method. can anyone solve this please...

Homework Statement Prove that the Frobenius norm is indeed a matrix norm. Homework Equations The definition of the the Frobenius norm is as follows: ||A||_F = sqrt{Ʃ(i=1..m)Ʃ(j=1..n)|A_ij|^2} The Attempt at a Solution I know that in order to prove that the Frobenius norm is indeed...

Homework Statement In Problems 25–30, x=0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x=0...

I'm reading up on some methods to solve differential equations. My textbook states the following: "y_{1} and y_{2} are linearly independent ... since \sigma_{1}-\sigma_2 is not an integer." Where y_{1} and y_{2} are the standard Frobenius series and \sigma_1 and \sigma_2 are the roots of...

My DE is \frac{h^2}{2m} \frac{d^2\psi}{dx^2} + \left(E - \frac{Ae^{-ax}}{x} \right) \psi = 0 where h, m, A < 0 and a and E are constants. I need to construct the following series solution (using the larger root of my indicial equation): \psi(x) = a_0 \left[x + \frac{Am}{h^2}x^2 +...

What conditions most be true for these two norms to be equal? Or are they always equal?

When using the Frobenius method of solving differential equations using power series solutions, I get a solution y = (indicial_stuff) + (infinite_summation_stuff) = 0 for a differential equation differential_stuff = 0. WHY is it that I can say (indicial_stuff) = 0? If y =...