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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How do I solve this nonlinear ODE given the asymptotic series solutions as follows?

Differential Equation: $$-{\frac { \left( {\frac {\rm d}{{\rm d}R}}f \left( R \right) \right) ^{2}}{2\,f \left( R \right) }}+{\frac {{\rm d}^{2}}{{\rm d}{R}^{2}}}f \left( R \right) +{\frac {{\frac {...
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Given $f(z) = |z|^{z_1}$, what is $ f'{(z)}$.

Context I am working on the series solution of a second-order, homogeneous, linear, ordinary differential equation. Irrespective of the two roots of the indical equation, one of the two linearly ...
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Bessel equation for half integer

Consider the Bessel differential equation $$x^2y''+xy'+(x^2-\mu^2)y=0$$. The indicial equation gives $r=\pm \mu$, then we have $a_1(1+2r)=0$, and then the recurrence is $$ a_k= \dfrac{a_{k-2}}{(k+r+\...
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Frobenius method solution

Suppose I have a second order linear differential equation. I have a solution about a regular singular point say $x=0$. Suppose the indicial equation has a repeated root for the indicial equation. I ...
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Difference between ODE solutions

Suppose I have a differential equation $$ x(1-x)y'' +8y' + 4y=0$$. Now suppose I start solving this for $x=2$ which is ordinary point. I choose a trial solution as $$y= \sum a_mx^m$$. Now now I solve ...
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Frobenius solutions

I have a second order linear ode which I am solving about a regular singular point $x=0$. The ode is $$ xy"+2y'+xy=0$$ I found the equation as $$ (r^2+r)a_0x^{r-1}+ (r^2+3r+2)a_1x^r +$$ the recurrence ...
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Frobenius method on Laplace equation in polar coordinates

After separating variables in a Laplace equation in polar coordinates, I have to solve the resulting Bessel equation for the $R$-variables (the $\Theta$ variable I do not consider in this post as it ...
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Can't seem to get a recurrence relation when applying the Frobenius Method to the following ODE.

$$x^2y'' -2xy' + 2y = 0 $$ Substituting in $ y = \sum_{i = 0}^\infty a_ix^i$ to obtain: $$ \sum_{i = 2}^\infty i(i-1)a_ix^i -2\sum_{i = 1}^\infty ia_ix^i +2\sum_{i = 0}^\infty a_ix^i = 0. $$ Matching ...
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Find 2 solutions of the Bessel equation as series of $x$

I'm trying to solve a problem from the textbook mathematics for physicist by Susan Lea and I have a few questions about it. First of all, I have to find 2 solutions as power of $x$ for the Bessel ...
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On Heun functions and Frobenius method when the characteristc exponents differ by an integer

Considere both equations below: \begin{align} y_{01}(z) &= H\ell(a,q;\alpha,\beta,\gamma,\delta;z)\\ y_{02}(z) &= z^{1-\gamma}H\ell(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha + 1 - \gamma,\beta + ...
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Understanding special functions

Tomorrow is my mathematical method exam where we have studied different kind of special functions named Legendre, Bessel's, Hermite and Laguerre functions. I solve their associated differential ...
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How to find first solution by Frobenius Method in that ODE?

My ODE is: y'' + (c-x)y = 0, where c is constant Using Frobenius Method: $y=\sum_{n=0}^{\infty}a_nx^{r+n}$ The equation is now $\sum_{n=0}^{\infty}a_n(r+n)(r+n-1)x^{r+n-2} + c\sum_{n=0}^{\infty}a_nx^{...
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Frobenius Method and Recurrence relations

Given the D.E: \begin{equation} z^2f''+p(z)zf'+q(z)f=0 \end{equation} We seek for solutions of the form $z^r\sum_{n=0}^{\infty}a_nz^n$, well-known as the Frobenius Method. Wikipedia gives the ...
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how to obtain Frobenius series solution to ODE when one term is multiplied by x to fractional power?

This is an ode from an old book, which I am stuck solving, since there is a term in front of $y''$ which does not have a power series. Here is the problem $$ x^{\frac{3}{2}} y'' + y = 0 $$ ...
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Differential Equations Power Series problem

The given equation is: $$x^2y''+xy'+\left(x-5\right)y=0$$ Using the method of frobenius where: $$y = \sum_{n=0}^\infty A_nx^{n+r}$$ $$y' = \sum_{n=0}^\infty A_n(n+r)x^{n+r-1}$$ $$y'' = \sum_{n=0}^\...
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Reference request : Fuchs' theorem for inhomogeneous equations

On the wikipedia page of Fuchs' theorem, the result is stated in the inhomogeneous setting. However, I was only able to find books which treat the homogeneous case (i.e. which talk about Frobenius ...
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How do you solve this inhomogenous differential equation by Frobenius method/Series solution?

So I'm solving some practice questions and I have a differential equation $$y''-3y'+2y=\sin(x)$$ about $x=0$, and I'm told to solve it using the Frobenius method. I don't think that's possible because ...
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Deriving the form of the second Frobenius solution when roots differ by an integer.

While using the Frobenius method to solve a second order ODE of the form $$y^{\prime \prime} + p(x)y^\prime + q(x)y = 0$$ if the roots of the indicial equation $(r(r-1)+p_0r+q_0=0)$ $r_1, r_2 (r_1>...
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Solutions to an ODE using Frobenius when roots are repeated

I have been looking at the following differential equation $$xy^{\prime \prime} + y^{\prime} + xy=0$$ Observe that since $x=0$ is a regular singular point we use the Frobenius method and subsutitute $...
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Duality functor is equivalence of category

I am reading a book on Frobenius algebra in which I read all the following definitions: Given a linear map $f:V\to W$ the dual map is defined by $f^{*}:W^{*}\to V^{*}$( as, $h\to f\circ h$) Let there ...
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Finding a second solution of $xy''+2xy'+6e^xy=0$

I am trying to solve $$xy''+2xy'+6e^xy=0,\space x>0$$ about $x=0$ using Frobenius method. The question is from Boyce & DiPrima. I have found the exponents at singularity to be $r_1=1$ and $r_2=...
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Frobenius Series solution to this ODE

I have an ODE of the form $$f^2 p^{\prime \prime}(r) + ff^{\prime}p^{\prime}(r) + V(r)p(r) = 0$$ where $f(r) = (1 - \frac{2M}{r} + \frac{Q^2}{r^2}) = \frac{1}{r^2}(r^2 - 2Mr + Q^2) = \frac{(r-r_+)(r-...
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Why does the smaller root of the indicial equation (in Frobenius method) not always yield a solution?

When using the Frobenius method, and the roots of the indicial equation differ by an integer, the larger one will always yield a solution, while the smaller may or may not. Is there a good way to ...
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Solving $x^2y'' - 5xy' + 6y = 0$ Through Power Series

I've been trying to solve the differential equation $$ x^2y'' - 5xy' + 6y = 0 $$ around the point $x_0 = 0$ through the Frobenius Method. I've already gotten the roots of the indicial equation ($x = 3 ...
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Comparing Summations & Frobenius Norms

Given that $x_i \in \mathbb{R}^d$,and$$m_1=\frac{1}{N}\sum\limits_{i}||x_i-\mu||_2^2$$$$m_2=\frac{1}{N^2}\sum\limits_{i}\sum\limits_{j}||x_i-x_j||_2^2$$ where $\mu=\frac{1}{N}\sum\limits_{i}x_i$ I am ...
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How to show following bessel equation property is true

Let $$x^{2}y'' + xy' + (\alpha^{2}x^{2}- n^{2})y = 0$$ by multiplying the above equation by $2xy'$ show that $$\int_{0}^{c} x(J_{0}(\alpha x))^{2} = \frac{c^{2}}{2}((J_{0}(\alpha c))^{2}+(J_{1}(\alpha ...
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A solution of the ODE $xy''=y$

Consider the ODE $$ xy''=y $$ in the interval $(0,\infty)$. By plugging $$ y=\sum_{n=0}^{\infty}a_nx^n $$ in the ODE, it is not hard to see that $a_0=0$ and $a_{n}=\frac{a_{n-1}}{(n-1)n}$ for all $2\...
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Frobenius Method for indicial equations but my powers aren't the same

I've been doing some work on the frobenius method and I've been able to successfully use it to obtain indicial equations and roots. However, in the this question I can't seem to make the powers equal ...
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Zero term in Frobenius series in derivation of Bessel functions.

When deriving Bessel functions by solving the Bessel equation \begin{equation} x^2y''+xy'+(x^2-n^2)y=0 \end{equation} using Frobenius method. In the resulting series \begin{equation} y = \sum\limits_{...
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Obtaining Indicial Equation

I think my question is pretty simple but I'm struggling to understand how to find indicial equations and would really appreciate some help. I have this equation: $$4xy''(x) +2y'(x) + y(x) = 0$$ I want ...
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I'm trying to solve a 2nd order ODE with frobenius method

I was trying to solve the ODE $$2x(1+x)y'' + (3+x)y' - xy = 0$$ but I couldn't seem to get the recurrence expression. I already got the roots of the indicial equation which are $r=0$ and $r=-1/2$ but ...
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Solve the following differential equation by undetermined coefficients $(1-x^2)y''-2xy'+6y=0$

I'm trying to obtain a recursion formula relating $a_{n+2}$ to $a_n$ using the method of undetermined coefficients. Then determining $a_n$ explicitly for each $n$ and finding the sum of the series. $$(...
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Frobenius Method solution and radius of convergence

I was given the equation $$y''-2xy'+\mu y = 0 $$ where $\mu$ is a parameter $\geq{0}$. I got the relation of sums: $$ \sum_{n=0}^{\infty} a_{n+2}(n+1)(n+2)x^{n} - 2\sum_{n=0}^{\infty} a_{n}nx^{n} + \...
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Finding Radius of Convergence from Frobenius Method

Given the equation $$y''-2xy'+\mu y=0$$ $P(x) = -2x$ and $Q(x) = \mu$ so $x_{0}=0$ is an ordinary point. I have the recurrence relation: $$a_{n+2}=\frac{a_{n}(\mu - 2n)}{(n+1)(n+2)}$$ With this I get ...
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Using method of Frobenius to find linearly independent solutions

I have been given the equation $$y'' -2xy' +(\mu)y = 0 $$ where $\mu$ is a fixed parameter $\geq0$. I solve it with $x_{0} = 0$. After plugging in $y = \sum_{n=0}^{\infty} a_{n}x^{n} $ into the ...
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Second Order Differential Equation - Two Irregular points

Consider the following second order differential equation $$ z^2 \psi''(z)+(z a+b) \psi'(z)+(b z+c) \psi(z) = 0 $$ where $a,b$ and $c$ are arbitrary real numbers. The equation has two irregular ...
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Frobenius Method on higher degrees

I would like to know if the Frobenius Method works on higher degree ODEs (3rd and so on). I tried to search some literature, articles etc on this matter, however I could not find anything except this: ...
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Solving Frobenius Method ODE Via Reduction Of Order

I am solving $x^2 y'' + 3xy' + (1 - 2x)y = 0$ using the Method of Frobenius. The indicial equation of this ODE has the solution $s = -1$ with algebraic multiplicity $2$. It is relatively easy to ...
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Frobenius method to solve $xy''+(1-x)y'-y=0$

Frobenius method to solve this equation: $$xy''+(1-x)y'-y=0$$ I am trying to find the solution for the DE but I'm stuck at the step where to find the equation to find the terms of the sequence....
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Relationship Between Power Series ODE Solution Techniques?

When solving an ODE via a power series at an ordinary (nonsingular) point, the initial guess is $y = \sum_{n = 0}^\infty a_n x^n$. When solving an Euler ODE, the second order equation $x^2 y'' + pxy' +...
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The Method Of Frobenius

The ODE $xy'' + y = 0$ has a real degeneracy. Use The Method Of Frobenius to find a fundamental set of solutions. Here is the procedure, as I understand it: 1) Plug the guess $y = x^s \sum_{n = 0}^\...
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Solving NH 2nd order ODE using Frobenius method

Quite stumped with this one so far. I have the following non-homogeneous ODE: $$2x^2y''+3xy'-xy=x^2+2x$$ And I need to find a solution for $x_0<0$ using Frobenius. Obviously we can center the ...
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Definition of regular singular point of ordinary differential equations.

I am studying the Frobenius method to solve series solution of differential equations around a regular singular point. I am using a book by an Indian author which defines the definitions as follows: ...
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Method of Frobenius Confusion

$x^2 y'' + 3xy' + (1 - 2x)y = 0$ I plugged in the guess $y = x^s \sum_{n = 0}^\infty a_n x^n$ to get $(s^2 + 2s + 1)a_0 x^s + \sum_{n = 0}^\infty (n^2 + 2ns + s^2 + 2n + 2s + 1)a_n x^{n + s} - 2a_{n - ...
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Reduction of Order On Functions

Suppose $y_1 (x)$ is a solution to $y'' + p(x)y' + q(x)y = 0$. Use reduction of order to show that a second solution is $y_2 (x) = y_1 (x) \int \frac{e^{-\int p(x)dx}}{(y_1 (x))^2} dx$. I plugged ...
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Frobenius theorem to solve PDE v.s. other techniques

When is the Frobenius theorem used to prove existence for PDE on manifolds, as opposed to more analytical techniques? I apologize that my question is pretty vague, but it stems from confusion about ...
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Method of Frobenius for a matrix differential equation?

Suppose we have the following matrix differential equation, $$\textbf{Y}'(x)=\frac{1}{x}\textbf{A}(x)\textbf{Y}(x),$$ where $$\textbf{Y}(x) = [y_1(x), y_2(x), ..., y_m(x)]^T$$ is a vector and $$\...
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Solving a linear ODE using Frobenius' method

The given ODE is $2y''$ - $y'/(x-1)$ + $y/(x-1)^2$ = 0. Solving we get, $\sum_{n=0}$$[2(n+r)(n+r-1)a_n - (n+r)a_n](x-1)^{(n+r-1)}$. Now I have few basic doubts. Please bear with me. Can't we directly ...
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Series Solution of ODE at a Regular Singular Point

I am given the ODE: $$ (\sin{x})y''+2(\cos{x})y'-(\sin{x})y=0 $$ and asked to: Show $x=0$ is a regular, singular point Show both linearly independent solutions can be written as a power series about $...
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Solving $(x-1)y'' -xy' +y = 0$ using Frobenius Series

I wish to solve $(x-1)y'' -xy' +y = 0$ using Frobenius series. The issue I am running into is the $x$ coefficient of the first derivative as I wish to find a solution centered at $x=1$. How would one ...
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