# 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 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 ...
1 vote
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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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### 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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1 vote
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### Zero term in Frobenius series in derivation of Bessel functions.

When deriving Bessel functions by solving the Bessel equation $$x^2y''+xy'+(x^2-n^2)y=0$$ using Frobenius method. In the resulting series 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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### 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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### 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 ...
1 vote
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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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