Intuitive explanation of Frobenius method

Question

There are some things that seem unralated and I find hard to even remember about Frobenius method because they don't even make sense to me. I've been studying power series to solve differential equations and everything went fine, I could find solutions of differential equations that can't be expressed on elementary functions. But then my book introduced the Frobenius method, apparently a enhanced power series method, it said that now I had to determine ordinary and singular points, all of this through the coefficients of the equations, but it never is explained why should I do that, Why all of the sudden coefficients can determine if a series converges or not? Why it matters if a singular point is regular or irregular? Why this singular points handicap the power series method?

Also it was introduced the indicial equation to determine r, which I saw arise from solving equations, when substituting the modified power series multiplied by x^r, when I had to find the coefficients values. But what I find weird is that such values, r1 and r2, determine the kind of solutions the equation has, Why the difference between r1 and r2 can determine that? Is it just a rule that seems to be obeyed or it has a mathematical proof or meaning?

Answer 1

This link might answer some of your questions. To answer your questions in my own words, consider the differential equation $$x^2y'' + xp(x)y'+q(x)y$$ to which we want to find a power series solution centered at $x=0$. Dividing by $x^2$ and renaming the functions $p(x)$ and $q(x)$, the equation becomes $$y'' + \frac{p(x)}{x}y'+\frac{q(x)}{x^2}y$$ Why do we require $p(x)/x$ and $q(x)/x^2$ to be a regular singular point at $x=0$? Since 'regular' means that we can express these two functions as power series around $0$. Only because of this fact can we make a power series ansatz for $y$ and then compare coefficients to fully determine $y$. If $p(x)/x$ and $q(x)/x^2$ weren't expressible as a power series, this obviously wouldn't work.

Then for your second question, the difference $r_1 -r_2$ indeed determines the solutions. This is theorem 2 in the link above. A reference to a proof is also given.

Answer 2

The "regular" domain of an ODE $y'=f(x,y)$ is a connected open set where $f$ is continuous. The more "regularity" you expect of a solution ("uniqueness" also being some kind of "regularity"), the more "regular" $f$ has to be. If you want power series expansions of the solution, you need that $f$ is analytical. This is what you said to be comfortable with.

Now it happens that one wants to extend solutions to the boundary of the regular domain and control the behavior there, for instance sift out the solutions that stay bounded. This for instance happens if one reduces a PDE on 3D-space via some spherical symmetry to an ODE in radial direction. Then the origin is usually a singular point, but the PDE solutions should not be singular there. Thus one needs solutions that are at least bounded at the origin.

There is one class of singular ODE that are treated early on, the Euler-Cauchy DE like $x^2y''+pxy'+qy=0$, $p,q$ constants. There the basis solutions that are powers of $x$, in special situations with a logarithmic factor, are easy to determine. One now can try to extend this situation by perturbations, obtaining a correction as perturbation series.

And indeed that is possible if the linear DE, with the singularity under examination shifted to the origin, can be written like $x^2y''(x)+p(x)xy'(x)+q(x)y(x)=r(x)$ where $p,q,r$ are at least continuous, but best analytical in $x=0$. Then if $x^r$ is a basis solution to $x^2y''(x)+p(0)xy'(x)+q(0)y(x)=0$, one can try to find a solution of the form $y(x)=x^ru(x)$ where $u$ is at least bounded in $x=0$, thus continuous there.

In the full analytical case that is assumed for the Frobenius series expansion, perturbation expansions become a little more complicated if there are resonances. In the base Euler-Cauchy case this means that $r=r_1=r_2$ is a double root of the characteristic equation. In the general case these can also occur for perturbation terms of a higher order/degree, that is, if $r_1+n=r_2$, where $n$ is an integer. Then $u$ is not simply a power series, it will also contain logarithmic terms. Thus the complicated statements about a general solution.

But anyway, since a general solution theory exists, with relatively "nice" solutions, this situation is called a "regular singularity".