# 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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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 $y(... 0answers 13 views ### Solving linear equations with alpha and beta (Frobenius method) Elements α, β ∈ R in the equations below. I would like to know how to solve these with Frobenius theorem, as I just recently started taking linear algebra courses I struggle with figuring this one out.... 0answers 10 views ### Why moving differentiation operator from outside to inside of ODE operator when solving for 2nd solution of an ODE w/ repeated indicial roots is valid So after solving for the first solution, we are advised to follow the method of isolating the$a_0$term involving the variable$r$from the indicial equation, but without substituting in its actual ... 1answer 49 views ### Is it possible to determine the end behavior of a power series based on the sequence of coefficients? Question: Given some sequence of coefficients (either explicitly or by way of a recursion formula) for a power series, is it possible to determine the end behavior (i.e. convergence, bounded, ... 1answer 81 views ### Polynomial solution$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 ... 0answers 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 ... 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 $$f_{(\lambda)} = \lVert D^{1-\lambda} U^* D^{2\lambda}\ UD^{... 0answers 48 views ### Special function related to a nonlinear ODE I am interested in finding a special function solution of the following ODE (r S)^{\prime \prime} = - 2 r S^2 with initial conditions S(0)=1, S^\prime(0) = 0 . The method of Frobenius gives ... 1answer 79 views ### Solving for the recursion relation of a second order ODE I want to solve for the asymptotic solution of the following differential equation$$ \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$$as y\... 1answer 113 views ### How to solve Legendre's differential equation without power series assumption? Legendre's differential equation \,(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+\ell(\ell+1)y=0\, is usually solved in most text-books either by assuming a power series solution or by Frobenius method.... 0answers 57 views ### Why will the \alpha I:d X^{-1}=- \alpha I:X^{-1}dX X^{-1}? Frobenius product A:B=Tr(A^TB) I see the calculation in this question : matrix multiplication ,which each element in this matrix is matrix too .\mathbf P\mathbf P^H = \frac{P}{t} \mathbf I But i ... 2answers 913 views ### Finding the solution to xy'' +2y' +xy=0 around x_{0}=0using the method of Frobenius. We know that the solution of this ODE is like:$$ 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^... 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, ... 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 = \...
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 ...