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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Using the Frobenius method for $ 4z^2 w''(z) - 4z^2 w'(z) + (1 - 2z) w(z) = 0$ around the point $z_0=0$.
Using the Frobenius method, find the solutions of the following differential equation in the complex plane around the point $z_0 = 0$:
$ 4z^2 w''(z) - 4z^2 w'(z) + (1 - 2z) w(z) = 0. $
(Hint: To ...
user1315539
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I am interested in formally verifying that $w_1(z) = 1 + \frac{1}{z}$ and $w_2(z) = \frac{1}{z} e^{-z}$ are linearly independent solutions.
When solving a differential equation in the complex space, I obtained the solution in the vicinity of $z_0 = 0$:
$ w(z) = A\left(1 + \frac{1}{z}\right) + B\left(\frac{1}{z} e^{-z}\right). $
Now I am ...
user1315539
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$ (1 + z^2)w'' - 2w = 0 $ in complex space
Find the linearly independent solutions of the equation $(1 + z^2)w''-2w = 0$
in the vicinity of the point $0$.
Attempt: To find the linearly independent solutions to the differential equation: $(1 + ...
user1333602
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Find the linearly independent solutions of the equation: $ z w'' + (3z - 1)w' - 3w = 0 $ in the vicinity of point 0.
Find the linearly independent solutions of the equation:
$
z w'' + (3z - 1)w' - 3w = 0
$
in the vicinity of point 0 and express them using elementary functions.
Attempt: Using the Frobenius method, we ...
user1315539
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Find the linearly independent solutions of the equation $ (1 + 4z^2)w'' + 16zw' + 8w = 0 $
Find the linearly independent solutions of the equation
$
(1 + 4z^2)w'' + 16zw' + 8w = 0
$
in the vicinity of point 0 and express them using elementary functions.
Attempt:
Let's solve the given ...
user1315539
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Deriving a three term recurrence relation from a DE using the Frobenius Method
I am trying to derive a three-term recurrence relation from the DE given by the equation below:
$$
\frac{d^2R(r)}{dr^2} + \frac{2}{r}\frac{dR(r)}{dr} + \left(2E - V(r) - \frac{l(l+1)}{r^2}\right)R(r) =...
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Find all solutions of the following DE in the vicinity of the point $z_0=0$: $z^2w''(z)+z(z+1)w'(z)-w(z)=0$ in the complex space.
We want to find all solutions of the following differential equation in the vicinity of the point $z_0 = 0$ in the complex plane:
$$ z^2 w''(z) + z(z+1)w'(z) - w(z) = 0 $$
Let's assume a solution in ...
user1315539
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Power series solution to $x^2y''+y'+y=0$ around $x=0$
If an ODE of the form
$$y''+p(x)y'+q(x)y=0$$
has $p(x)$ and $q(x)$ that are analytic at $x_0$ then we can suppose a power series solution as such:
$$y(x)=\sum_{n\geq 0} a_n(x-x_0)^n$$
If instead they ...
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Second particular solution to $xy''+y'-xy=0$ [duplicate]
Assuming $y=\sum_{n\geq 0}a_nx^{n+r}$ (Fröbenius method), $xy''+y'-xy=0$ reduces to
$$\sum_{n\geq 0}a_n(n+r)(n+r-1)x^{n+r-1}+\sum_{n\geq 0}a_n(n+r)x^{n+r-1}-\sum_{n\geq 0}a_nx^{n+r+1}=0$$
Reindexing ...
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Solving $x^2y''+2xy'+2y=0$ using power series
One way to solve $x^2y''+2xy'+2y=0$ is using the substitution $x=e^t$. This time I'm asked to use power series (Fröbenius method). Assuming $y=\sum_{n\geq 0}a_nx^{n+r}$, the diff eq reduces to
$$\sum_{...
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Frobenius method, cant find the roots
I am trying to solve this equation with Frobenius method :
$(x^2-x)y''(x) + (1/3) y'(x) + xe^{(x-1)}y(x)=0$
I am trying to find the roots with : $r^2 + (P_0-1) + Q_0 = 0$
And, $P_0 = (1/0!)(1/3x)^{(0)}...
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Power Series and Frobenius, when i use on this question?
I have a question like this
Find the power series solution of the equation $y'' + ty' + 3y = 0$ near the point $t_0=0$
When I solve this problem I couldn't see easily which is $P(x), Q(x)$ in this ...
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Finding a complicated recursion relation by Frobenius' method
I am trying to find quasinormal modes of Kerr black holes, following https://www.edleaver.com/Misc/EdLeaver/Publications/AnalyticRepresentationForQuasinormalModesOfKerrBlackHoles.pdf. To do this, I ...
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Regular singular points for differential equation systems
Suppose that we are given a first-order system of ordinary differential equations
$$\frac{\mathrm{d} f(x)}{\mathrm{d}x} = A(x) f(x),$$
where $A(x)$ is an $N \times N$ matrix which has an expansion of ...
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Frobenius solution with non-patterned recurrence relation
So I'm trying to solve the following ODE which have three ordinary singular points:
$$z(z-1)(z-\tau)R''(z)+(a+bz+cz^2+dz^3)R'(z)+(\alpha+\beta z+\gamma z^2)R(z)=0;$$
Using the Frobenius method, $R(z)=\...
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