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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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Frobenius series solutions and asymptotic

Consider the equation below which is an eigenvalue problem I am studying: $$ \label{left} \lambda(v-v'')+c\left(v''-v+be^\xi v+(1-b)e^\xi v'-e^\xi v''\right)'=0. $$ Restriction on the parameters: $\...
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Indical equation of Frobenius Method

So I'm getting ready for the exam by doing last years exam. It gives a DE which goes as following $$(x^2-x)y''+(4x-2)y'+2y=0$$ This gives me: $$\sum^\infty_{n=0}(n+r-1)(n+r)a_nx^{n+r}-\sum^\infty_{...
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Don't understand a result of a Frobenius method solution

So after using the Frobenius method on $2xy''-y'-y=0$ I get one of the results for y to be a series of the form $1+\sum_{n=1}^\infty \frac{x^n}{5\cdot7\cdot9\cdot\cdot\cdot(2n+3)n!} $ but the solution ...
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40 views

Frobenius series

Consider the differential equation \begin{equation} (1+x^3)y''+4xy'+y =0 \end{equation} I need to find a lower bound on the radius of convergence for above equation at $x=0\ \&\ x=2$. I wrote ...
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When using the Frobenius method, and r1-r2 is neither zero nor a positive integer, can you use the Wronskian to find the second solution?

When using the Frobenius method, and r1-r2 is neither zero nor a positive integer, can you use the Wronskian to find the second solution? Basically, do I have to repeat the substitution with the ...
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44 views

Recurrence relation for the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation $$ \left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$ as $y\...
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48 views

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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29 views

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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37 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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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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35 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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32 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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129 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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96 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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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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42 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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37 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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303 views

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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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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137 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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30 views

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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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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56 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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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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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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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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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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51 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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56 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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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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98 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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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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625 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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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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118 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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402 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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148 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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57 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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94 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
467 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
209 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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247 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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107 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
180 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 ...