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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28 views

Resolve the equation: $x^2 (1+2x) y''+ 2x(1+6x) y'-2y =0$ Using Frobenius method case 3 [closed]

Resolve the equation: $x^2 (1+2x) y''+ 2x(1+6x) y'-2y =0$ using method of Frobenius case 3.
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13 views

How do I use an indicial equation to show solutions of ODE? (Frobenius Method)

I'm given this equation:$$xy''+(1-x)y'+\alpha y=0$$, within the indicial equation $$r(r-1)+b_0 r+ c_0 =0$$ I don't understand how should I use the indicial equation to show the two solutions of this ...
3
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2answers
52 views

Use Frobenius' method to find two independent solutions to the ODE $4tx''(t)+2x'(t)+x(t)=0$

It is known that $t=0$ is a regular singular point of $4tx''(t)+2x'(t)+x(t)=0$. By Frobenius' method, show that two independent solutions of the ODE are given by $x_1(t)=\sum_{k=0}^\infty \frac{(-1)...
5
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1answer
237 views

Solve $x(1-x)y''+2(1-2x)y'-2y=0$ by the Frobenius Method

Find the second solution. First solution is $\dfrac 1 {1-x}$. Solve by Frobenius Method: $$x(1-x)y''+2(1-2x)y'-2y=0\,.$$ The first solution I am able to get is $\dfrac{1}{1-x}$. Other solution is $\...
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0answers
42 views

Method of Frobenius with $\lambda_1-\lambda_2 \in \mathbb{N}^+$, $\lambda_1>\lambda_2$

When working with series solutions near a regular singular point, we apply the method of Frobenius, with at least one solution of the form $$y=x^{\lambda}\sum^{\infty}_{n=0}a_nx^n \tag{1}$$ ...
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1answer
30 views

Find $y_1(x)$ and $y_2(x)$ from the recurrence relation of $r_2$ only, by using the Frobenius method

The given original equation is: $$x^2y^{''}+xy^{'}+(x^2-\frac{1}{4})y=0$$ The series I have is: $$x^r\left[\left(r^2-r+r+x^2-\frac{1}{4}\right)C_0+\left(r^2+r+1+r+x^2-\frac{1}{4} \right)C_1x\right]+\...
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1answer
51 views

What is the meaning of “perhaps” here? “[The solution] is valid in the common interval of convergence … except perhaps for $x=x_0$ …”

I am unclear about why won't a Frobenius series solution be valid at $x=x_0$ or why is it uncertain at $x=x_0$? In the theorem it is stated as "except perhaps at $x=x_0$". I haven't studied real ...
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2answers
16 views

How can I see the three solutions I created from Frobenius Solution are indeed linear dependent? (Or do they have to?)

Question: Compute the Frobenius Series about $x=1$ for the following problem: $(x-1)y"(x)-xy'(x)+y(x)=0$ For those who are not familiar about Frobenius Method, we basically try to compute $y(x)=\...
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1answer
71 views

The second solution for $xy''+y=0$.

Solve $xy''+y=0$. After obtaining the first solution $y_1(x)$ using the Frobenius series $$y_1(x) = a_0\sum_{n=0}^\infty{\frac{(-1)^n}{n!(n+1)!}x^{n+1}}$$ I need to find the second solution which ...
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26 views

Different method for solving Legendre differential equation

Here is the Legendre differential equation: $$ (1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+l(l+1)y=0$$ It's well known that it can be solved using power series method but I wonder if there is another ...
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1answer
38 views

Frobenius solution of $xy"+y=0$

$$xy"+y=0$$ Since $x=0$ is singular point: $y(x)=\sum_{n=0}^\infty a_nx^{n+r}$ $$\implies \sum_{n=0}^\infty a_n(n+r)(n+r-1)x^{n+r-1}+\sum_{n=0}^\infty a_nx^{n+r}=0$$ How to proceed further?
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1answer
112 views

Lipschitz constant of negative log likelihood function

Show that $∇l(\mathbf{w})$ is Lipschitz continuous. I found that $∇l(\mathbf{w})$ is: So from the hint, Lipschitz constant is $\frac{1}{2}\frac{y_i}{y_i}||xi||$. I am not sure how to get the ...
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22 views

Can Frobenius method method be used to solve following ODE

I'm trying to solve following ODE $$\frac{d^2y}{dx^2}+y\tan^{-1}(x)=0$$ But the power series of $\tan^{-1}(x)$ is actually divided in there regions here. Moreover, this ODE doesn't have the standard ...
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40 views

Frobenius method giving y=0

I'm trying to solve $$y^{''}+\bigg(-\frac{1}{x}+\frac{1}{x^3}\bigg)y=0$$ using Frobenius method. What I did is following: $$y=\sum_{i=0}a_ix^{i+k}$$ $$y^{''}=\sum_{i=0}(i+k+2)(i+k+1)a_{i+2}x^{i+k}$$ ...
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1answer
32 views

Does Frobenius Norm affect matrix transpose?

On page 11 of the slide, Sum-of-least-square loss: $$ \ell\left(\mathbf{\tilde W}\right) = \sum_{n=1}^N \left\| \mathbf{\tilde W}^T\mathbf{\tilde x^{(n)}} -\mathbf{t}^{(n}) ...
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1answer
58 views

Does the Frobenius method work for all second order linear differential equations with only regular singular points?

For $$x^2y''+x(x^2+1)y'+(x-4)y=0,\tag1$$ there is a regular singular point at $0$, but when I tried to use the Frobenius method and substituted $$y=\sum_{n=0}^\infty a_n x^\left(n+r\right)\tag2$$ into ...
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1answer
37 views

To differentiate between normal power series solution and Frobenius Method

So I have learned about power series solution and Frobenius method in my Engineering Maths course in University, but i am quite confused about when to use which method to solve the 2nd order ...
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1answer
93 views

Can’t see that an ODE is equivalent to a Bessel equation

I can solve the following differential equation without any trouble using the method of Frobenius $$ x^2 y’’ - (2 + 3x) y = 0. $$ When I put the differential equation in Mathematica, it gives me the ...
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34 views

Frobenius method case II

I am solving the question xy''+(1-x)y'-y=0 This was exactly how I solved the question using y2= integral (e^-integral p(x)dx / y1^2 )dx but since this question involves repeated real roots I thought ...
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1answer
48 views

Frobenius method $ 2xy''-(3+2x)y'+y=0$

I tried solving $2xy''-(3+2x)y'+y=0$ using the Frobenius method, and I kept getting incorrect answers. I got a final form which is $$a_{k+1} = \frac{(-2k-2c+1)a_k }{ 2k^2+2c^2+4kc-c-k-3}$$ this ...
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1answer
43 views

Find a value of alpha such that all solutions of the differential equation exist

I was recently challenged to solve a question that is as follows: Determine the two values of the constant $\alpha$ for which all solutions of $xy'' + (x-1)y' - \alpha y = 0$ can be written as $y(...
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13 views

Solving linear equations with alpha and beta (Frobenius method)

Elements α, β ∈ R in the equations below. I would like to know how to solve these with Frobenius theorem, as I just recently started taking linear algebra courses I struggle with figuring this one out....
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10 views

Why moving differentiation operator from outside to inside of ODE operator when solving for 2nd solution of an ODE w/ repeated indicial roots is valid

So after solving for the first solution, we are advised to follow the method of isolating the $a_0$ term involving the variable $r$ from the indicial equation, but without substituting in its actual ...
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1answer
49 views

Is it possible to determine the end behavior of a power series based on the sequence of coefficients?

Question: Given some sequence of coefficients (either explicitly or by way of a recursion formula) for a power series, is it possible to determine the end behavior (i.e. convergence, bounded, ...
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1answer
81 views

Polynomial solution $xy''+(1-x)y'+ \lambda y=0$

For which values of the constant $\lambda$ does the differential equation $$xy''+(1-x)y'+ \lambda y=0$$ have a polynomial solution? I was thinking about solving this problem with the Theorem of ...
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35 views

Frobenius Method approach

How do we decide that in frobenius method, what kind of power series to take? One with increasing power or one with decreasing power? I know both leads to same answer in the end, but I wanted to know ...
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2answers
82 views

The differential of $\lVert D^{1-\lambda} U^* D^{2\lambda}\ UD^{1-\lambda} \rVert_F^2$ with respect to $\lambda$

Let a square matrix $A=WDV^*$ by the SVD where $D$ is diagonal with positive entries, $U=V^*W$ is unitary, and $0<\lambda<1$. let $$ f_{(\lambda)} = \lVert D^{1-\lambda} U^* D^{2\lambda}\ UD^{...
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48 views

Special function related to a nonlinear ODE

I am interested in finding a special function solution of the following ODE $ (r S)^{\prime \prime} = - 2 r S^2 $ with initial conditions $ S(0)=1, S^\prime(0) = 0 $. The method of Frobenius gives ...
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1answer
79 views

Solving for the recursion relation of a second order ODE

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

How to solve Legendre's differential equation without power series assumption?

Legendre's differential equation $\,(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+\ell(\ell+1)y=0\, $ is usually solved in most text-books either by assuming a power series solution or by Frobenius method....
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57 views

Why will the $\alpha I:d X^{-1}=- \alpha I:X^{-1}dX X^{-1}$?

Frobenius product $A:B=Tr(A^TB)$ I see the calculation in this question : matrix multiplication ,which each element in this matrix is matrix too .$\mathbf P\mathbf P^H = \frac{P}{t} \mathbf I$ But i ...
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2answers
913 views

Finding the solution to $xy'' +2y' +xy=0$ around $x_{0}=0$using the method of Frobenius.

We know that the solution of this ODE is like: $$ y=\sum_{n=0}^{\infty}C_nx^{n+r}$$ Them derivative $y$ and $y'$. $$y'=\sum_{n=0}^{\infty}(n+r)C_nx^{n+r-1}$$ $$y''=\sum_{n=0}^{\infty}(n+r-1)(n+r)C_nx^...
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1answer
66 views

Euler (equidimensional) equation question

Consider the equation $$x^2y''-8xy'+20y=0.$$ From an undergraduate ODE course, it is known that the two linearly are $y_1=x^5$ and $y_2=x^4$. However, why don't we consider solutions, for example, ...
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1answer
63 views

How do you arrive at this solution to the modified Bessel equation using the Frobenius method?

I'm looking to solve the equation $$ x^2y'' + xy' - (x^2+p^2)y = 0 $$ in particular for $p = \frac12$. I've already determined the indicial equation to be $r^2 - p^2 = 0$, and so the roots are $r = \...
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2answers
66 views

Finding indicial equation

I am given a differential equation: $$xy'' - 4y' + 5xy = 0$$ I am told that this has a singular point at $x=0$. I computed: \begin{align*}x p(x) &= -4\\ x^2 q(x) &= 5x^2 \end{align*} From ...
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2answers
86 views

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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1answer
30 views

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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1answer
24 views

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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2answers
63 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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0answers
29 views

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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1answer
66 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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0answers
56 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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2answers
50 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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1answer
47 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
47 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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1answer
41 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
43 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 ...
2
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1answer
245 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
58 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 ...
3
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1answer
75 views

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, ...