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Questions tagged [frobenius-method]

Use this tag when you want to solve a linear ordinary differential equation with variable coefficients via the Frobenius method.

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Solving ODE with Frobenius Theorem [on hold]

Question: Solve $(x-2)y^{''}+2xy=0$, given that $x=2$ is a singular point, and $r=1$. I have trouble while dealing with the $2x$. Anyone can kindly show me the whole solution for this question? :)
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Find the first Frobenius series solution with nonzero positive exponent (when $r=1$) of singularity up to four nonzero terms

The differential equation is $(x-2)y"+2xy=0$ with singular point $x = 2$. I'm all good until I got stuck at the recurrence relation part where i have an, an-1 and an-2 .. appreciate if someone can ...
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2answers
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Frobenius method on ODE; series expansion

$$x(x-1)y''+3xy'+y=0$$ $$y''+\frac{3x}{x-1}y''+\frac{1}{x(x-1)}y=0$$ So, this eq. has irregular points at $x=1$ and $x=0$; Using Frobenius method I can expand this thing arround $x=0$ anyway. As ...
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1answer
33 views

Solving a differential equation using the Frobenius method

I want to solve the differential equation $2ty'' + (1 - 2t)y' - y =0$ using Frobenius's method. I understand the method, and I've looked up several examples; however, I can't manage to solve this ...
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1answer
36 views

Frobenius Method: derivatives of y

http://mathworld.wolfram.com/FrobeniusMethod.html Was reading this and was wondering why the $n$ does not increase while computing the derivatives of $y$ in the frobenius method. In the first ...
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33 views

Series solution of the second order ODE around a regular singular point

Here is the ODE I want to integrate, $$R''(y)-\frac{2}{k-y}R'(y)-\frac{l(l+1)}{(k-y)^{2}}R(y)=0$$ We see that it has a regular singular point at $y=k$ where $k<0$. Is there a way to obtain the ...
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1answer
30 views

How can one solve the recurrence relation $a(n+3) = Ba(n)/n^2$?

As the title suggests, I am looking for the solutions to the recurrence relation $a(n+3) = B \frac{a(n)}{n^2}$. In particular, I am attempting to solve a differential equation using the power ...
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55 views

Frobenius norm and singular values

I study about random projection and i m really confuse about the relationship between Frobenius norm and singular values. The book say that the $||M||_f^2 $ and $\sigma$ had a correlation. I found ...
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1answer
50 views

Using Frobenius method to solve the Legendre differential equation

I'm tasked with solving the Legendre differential equation, and Using $c=0$, obtain a series of even powers of $x$ (with $a_1=0$). I found this exercise to be good at highlighting what I found ...
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1answer
38 views

How to calculate the $\frac{\partial det(\mathbf X)}{\partial \mathbf X}$ and $\frac{\partial tr(\mathbf X^n)}{\partial \mathbf X}$

How to calculate the $\frac{\partial det(\mathbf X)}{\partial \mathbf X}$ and $\frac{\partial tr(\mathbf X^n)}{\partial \mathbf X}$ by using Frobenius product?i tried to begin the calculation,but i ...
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Cannot solve recursion relation in power series solution to this ODE

I'm trying to solve the differential equation $$\frac{d^2u}{dr^2} - \left[V_0(r-1)^2 + \frac{\ell(\ell+1)}{r^2} \right]u = -\lambda u$$ where $r\geq0$ is the radial component in spherical coordinates, ...
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1answer
41 views

Establish radius of convergence by ratio test

I was able to solve the following : $(1-x^2)y''-2xy'+\lambda y = 0$ where $\lambda$ is a real constant. The solution that I obtained is of the form $\sum_{n=0}^{\infty}C_nx^n$ with $C_{n+2} = \...
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1answer
36 views

Solve second order differential equation with cosh using frobenios method

i need to show that the differential equation $y^{''}+(\cosh(2x)-4)y = 0$ has the solution: $ y(x) = x+\frac{1}{2}x^3-\frac{1}{40}x^5 -... $ using Frobenius method. I started by writing cosh(2x) ...
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How is $s^2+(q_0-1)(s+r_0)$ obtained in Frobenius method proof in (Special Functions for Scientists and Engineers)?

In Special Functions for Scientists and Engineers's very first section in the proof of Frobenius method, the equation (1.7) $a_0s(s-1)+a_0q_0s+a_0r_0=0$ is obtained as a special case for $i=0$ in the ...
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Roots of the Indicial Polynomial of the Legendre equation $(1-z^2)u''-2zu'+v(v+1)u=0$

Consider the Legendre equation $$(1-z^2)u''-2zu'+v(v+1)u=0.$$ Find the roots of the indicial polynomial if we apply the Frobenius method about $z=1$. My attempt: Let \begin{align}u=\sum_{k=0}^{\...
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1answer
46 views

Constructing a Bounded Solution using Frobenius' Method

For the ODE $$3z^2u''+8zu'+(z-2)u=0$$ construct a series solution of the form $$\sum_{k=0}^{\infty} A_kz^{k+r}$$ that is bounded as $z\rightarrow 0$. Take $A_0=1$ and compute explicitly the terms ...
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1answer
81 views

Why do I have to use Frobenius method in Bessel's equation?

I just learnt how to apply the power series method in differential equations and I'm now trying to understand the extended method of Frobenius. As an example, the textbook gives me Bessel's equation ...
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Evaluation of $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}n^2}{n^3+1}$ [duplicate]

I have accrossed this sum :$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}n^2}{n^3+1}$ in my textbook , However i have tried to evaluate it using Frobenius method and also the standard sum $\sum_{n=1}^{\infty} ...
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Frobenius norm of two related matrices

Given a matrix A, what is the relationship between the Frobenius norm of $A^TA$ and $A^TA - I$, where $I$ is the identity matrix. By relationship, I mean whether we can infer if one is bigger/smaller ...
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1answer
31 views

How does Frobenius method select two independent solutions?

Consider the ODE $$y'' - y = 0$$ with solutions $y_{(1)} = c_1 \cosh x + c_2 \sinh x$, or, equivalently, $y_{(2)} = c_1 \exp (+x) + c_2 \exp (-x)$. If we were to solve the above ODE by Frobenius ...
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Is the Method of Frobenius Appropriate for this DE? If so, how to proceed?

To not bore with motivation, I'll get straight to the point. I have struggled for quite some time finding analytical solutions to the differential equation $$y'' + \frac{a}{x}y' - \frac{b}{x^2\left(1 -...
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1answer
53 views

1-form on a differentiable manifold

I have a question in differential manifolds, If i have $\alpha$ a smooth 1-form on a differential manifold $M$ and we have that it is closed, and it doesn't vanish at any point in $M$. We have a ...
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89 views

Convex Quadratic Programming Problem with Frobenius Norm

How do I solve the optimization problem: $$ \begin{array}{rl} \min&\big\|w\big(\sum_{j=1}^L\alpha_jy_jx_j^\text{T}\big)\big\|_F^2\\ \text{s.t.}&0\leq\alpha_i\leq1\text{ for all }i=1,\dots,L \...
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53 views

Solution to a convex Quadratic Programming problem with Frobenius Norm Constraint

I have the following optimization problem: \begin{array}{l}\mathop {\min }\limits_{A \in {\mathbb{R} ^{d \times d}}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ...
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1answer
670 views

The formula for the $n^\text{th}$ term of $\frac{x^2}6 -\frac{x^4}9 + \frac{3x^6}{80} - \frac{71 x^8}{15120} + \frac{ 10361 x^{10}}{10886400} \dots$?

I obtained an infinite series after solving a non-linear differential equation using Frobenius method. It is possible to obtain the coefficient for an arbitrary power of the variable, but also time ...
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Extracting singular points from an equation of this form $y'' - y' + (a-x^2) y = 0$

I put this equation in \begin{equation} y'' - y' - (a-x^2)y=0 \end{equation} into Wolframapha, and it gave a linear combination of two solutions; the first was a Hermite polynomial solution and the ...
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119 views

Frobenius method roots differing by integer

$$ \ddot y(z) + a(z) \dot y(z) + b(z)y(z)=0 \ \color{red}{(1)}$$ where $a(z),b(z)$ are analytic on a neighborhood of z=0 (but not including this point) with Laurent serie: $$ a(z)=\sum_{k \ge 0} a_kz^{...
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1answer
54 views

Finding closed form of Frobenius solution

I've done the following: Consider the differential equation given by $\displaystyle y^{\prime\prime} - 2y^\prime + \frac{4x^2 + 1}{4x^2} y = 0$. As one can see immediately, $x=0$ is a singular. Is it ...
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Frobenius method and series solutions

I have understood how to apply the series solutions and the Frobenius method to solve second order ODE's of the form: $$y''+p(x)y'+q(x)y=0$$ I don't understand how $p(x)$ and/or $q(x)$ not being ...
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1answer
49 views

Solving differential equation using Frobenius Method

I've been given the problem to solve the following differential equation \begin{equation} x^2y''+(2x+3x^2)y'-2y=0 \end{equation} using Frobenius Method around the regular singular point $x=0$. From ...
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Frobenius Method Shortcut

Is there a shortcut for Frobenius method using the indicial equation or any other equation? I think I may have seen a shortcut to differentiating numerous sigma equations, but I cannot recall what it ...
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1answer
54 views

Why divide a second order differential equation, don't we lose solutions?

I'm learning to get series solutions for differential equations. My book says: If you have a second order differential equation of the type: $a_1(x) y'' + a_2(x)y' + a_3(x)y = 0$ we should first ...
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84 views

Second Order D.E with non-constant (trig-functions) coefficients

This is not a homework exercise. I am just making it clear. I have the following second-order differential equation. $$\frac{4 q'(z)^3 \sin ^3(q(z))}{z^2 \left(z^2 q'(z)^2+1\right)^{3/2}}+\frac{3 \...
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1answer
92 views

Frobenius Method - Non integer powers of $x$ in differential equation?

I am trying to solve an ODE using the Frobenius method. I understand the general process, but I do not understand how you compare coefficients when you have a $x^\frac{1}{2}$ term in the differential ...
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Series solution of a 2nd order ODE

Is the ODE $(1-x^2)y''+y'+y=0$ solvable by simple power series (not Frobenius) method? The reason I am asking this, is because if the eq were $(1-x^2)y''+xy'+y=0$, it could have been easy, since all ...
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1answer
37 views

Frobenius series problem

I have: $$4x^2y'' + (3x+1)y = 0$$ From this the indicial equation is found so that $$r(r-1)+1/4=0 \Rightarrow r = 1/2 = r_1 = r_2,$$ so the first solution will be of the form: $$y_1(x) = \sqrt{x}\...
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How can I solve $y'' + \left(4x-\tfrac{2}{x}\right)y' + 4x^2y= 3xe^{x^2}$?

The DE is $$y'' + \left(4x-\frac{2}{x}\right)y' + 4x^2y= 3xe^{x^2}.$$ I've been told to use $t = x^2$ along with change of variable to solve it, but it's clear that's not possible due to the ...
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1answer
519 views

Indicial equation of $(x^2-1)^2y''+(x+1)y'-y=0$

Let $\alpha$ and $\beta$ with $\alpha>\beta$ be the roots of the indicial equation of $(x^2-1)^2y''+(x+1)y'-y=0$ at $x= -1$. Then what is the value of $\alpha-4\beta$ ? I am trying to solve ...
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63 views

Solving a homogenous second order recurrence with polynomial coefficients

How would I find a closed-form solution of a recurrence that I obtained from solving an ODE using Frobenius method?: $(k+2)(k + 4)a_{k+2} - a_{k+1} + (k+2)a_k = 0$ given $a_0 = a$ and $a_1 = \frac{1}...
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1answer
102 views

Second order derivative of the squared Frobenius norm

I have a matrix $X$ of size $k\times d$. $k$ might be constrained to be equal to $d$. I'm searching for the derivative of the following equation with respect to $X$: $\Vert X^TX\Vert_F^2$ I tried ...
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1answer
353 views

Dealing with a large Kronecker product in Matlab

Following is my minimization problem to solve for matrix D: $$P=BDB^{T}$$ where dimension of $B$ is $576 \times 1296$ and dimension of $P$(unsymmetric) is $576\times 576$ and $D$ is a diagonal ...
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1answer
147 views

Does curl vector influence the final destination of a particle?

If we have an $n$ dimensional space ($n>3$) with a continuous $n$-dimensional vector field $\boldsymbol F$ $$\boldsymbol F:\mathbb{R}^n\rightarrow\mathbb{R}^n$$ and for every particle in this ...
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1answer
55 views

Writing a product as a factorial

So I'm solving a differential equation with the Frobenius method but I'm stuck with the coefficients. I get that $a_n = \frac{(-2)^na_0}{n!(3n+1)(3n-2)(3n-5)...1}$ but I can't figure out how to write ...
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1answer
85 views

Two different recurrent relations for the same index using frobenius' method

I'm trying to solve the following differential equation using Frobenius' method: 4xy''+2y''+y=0 I checked x=0 is a regular singular point and performed the substitution $y=x^{\lambda}\sum_{n=0}^{\...
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1answer
401 views

Irregular singular point & Frobenius form

I have got the following differential equation. $$z^3{\rm y}′′(z) + z\,{\rm y}′(z) + \lambda\,{\rm y} (z)=0$$ where $\lambda$ is a constant. I think I proved that at $z=0,$ the ODE has an ...
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1answer
205 views

Frobenius norm derivative

What is the partial derivative of the following expression with respect to T: $‖X−PTKV‖^2$ Where ∥.∥ denotes the Frobenius norm, and X,P, T , K and V are matrices. thanks.
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1answer
236 views

Derivative of norm

What is the partial derivative of this expression with respect to T: $$‖X−PTV‖^2$$ Where ∥.∥ is the Frobenius norm, and X,P, T and V are matrices. thanks.
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1answer
105 views

$ y''\sin^2 x + y' \tan x + y \cos^2x = 0 $

I have been going insane over the following differential equation over the past few days. $y''\sin^2(x) + y'\tan(x) + y\cos^2(x) = 0 $ The assignment is: $a)$ Show that $x=0$ is a ...
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1answer
155 views

Frobenius Method for r = 1

The general methodology for this involves assuming a solution of the form $$ y = \sum_{n=0}^\infty a_nx^{n+r}.$$ One normally keeps the index $0$ for the first and second derivatives. My question ...
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1answer
49 views

Figuring out the equivalent power series based on pattern

I am trying out to solve a differential equation using a power series solution. I figured out the recurrence or recursive relation of the constants and I figured out the pattern somehow. By ...