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Am I correct in believing Frobenius' method is simply a general case of a power series solution? For instance, if the Differential Equation has a regular singular point, would you be forced to use Frobenius' method instead of a regular power series?

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In fact Frobenius method is just an extension from the power series method that you add an additional power that may not be an integer to each term in a power series or even add the log term for the assumptions of the solution form of the linear ODEs so that you can find all groups of the linearly independent solutions that in cases of cannot find all groups of the linearly independent solutions when using power series method.

You can force to use Frobenius method when you find that the linear ODEs can already find all groups of the linearly independent solutions when using power series method, however, you will find that you can already find all groups of the linearly independent solutions when the additional power is just taking an non-negative integer and no need to add the log term.

If you doubt that whether the linear ODEs can find all groups of the linearly independent solutions when using power series method or not, you can feel free to use Frobenius method instead. The only thing is you should write more steps.

For example you force to solve $y''+xy=0$ by Frobenius method:

Let $y=\sum\limits_{n=0}^\infty a_nx^{n+r}$ ,

Then $y'=\sum\limits_{n=0}^\infty(n+r)a_nx^{n+r-1}$

$y''=\sum\limits_{n=0}^\infty(n+r)(n+r-1)a_nx^{n+r-2}$

$\therefore\sum\limits_{n=0}^\infty(n+r)(n+r-1)a_nx^{n+r-2}+x\sum\limits_{n=0}^\infty a_nx^{n+r}=0$

$\sum\limits_{n=0}^\infty(n+r)(n+r-1)a_nx^{n+r-2}+\sum\limits_{n=0}^\infty a_nx^{n+r+1}=0$

$\sum\limits_{n=0}^\infty(n+r)(n+r-1)a_nx^{n+r-2}+\sum\limits_{n=3}^\infty a_{n-3}x^{n+r-2}=0$

$r(r-1)a_0x^{r-2}+r(r+1)a_1x^{r-1}+(r+1)(r+2)a_2x^r+\sum\limits_{n=3}^\infty((n+r)(n+r-1)a_n+a_{n-3})x^{n+r-2}=0$

$\therefore r=1,0,-1,-2$

When we take $r=0$ ,

$2a_2+\sum\limits_{n=3}^\infty(n(n-1)a_n+a_{n-3})x^{n-2}=0$

As we can already find all groups of the linearly independent solutions from this relation when we take $r$ as non-negative integer, and when taking $r$ as non-negative integer, the assumptions of the solution form is as same as assuming the solution form as power series. This implies that we can solve $y''+xy=0$ by power series method.

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See this link: http://en.wikipedia.org/wiki/Frobenius_method

You are correct. For Froebenius, you add an additional power that may not be an integer to each term in a power series, and use a 2nd order differential equation + initial conditions to determine that additional power. In many cases, you solve a quadratic equation for that additional power. If there is a double root, then there is a logarithm in the series solution. (See the example in the link; other examples include the BesselY function.)

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