# Questions tagged [frobenius-method]

Use this tag when you want to solve a linear ordinary differential equation with variable coefficients via the Frobenius method.

109 questions
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### Solving ODE with Frobenius Theorem [on hold]

Question: Solve $(x-2)y^{''}+2xy=0$, given that $x=2$ is a singular point, and $r=1$. I have trouble while dealing with the $2x$. Anyone can kindly show me the whole solution for this question? :)
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Extracting singular points from an equation of this form $y'' - y' + (a-x^2) y = 0$

I put this equation in \begin{equation} y'' - y' - (a-x^2)y=0 \end{equation} into Wolframapha, and it gave a linear combination of two solutions; the first was a Hermite polynomial solution and the ...
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### 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 ...