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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, like the following one: $$ y=\left\{ \begin{array}{c} x^5\;\;{\mbox{if}}\;\;x\leq 0\\ 0\;\;{\mbox{if}}\;\;x> 0. \end{array} \right. $$ While the solution above is only differentiable 4 times, it is a perfectly good solution to the equation.

I understand that the point $x=0$ is singular and thus the uniqueness theorem does not apply there. Also this is not unique to that equation or that solution. The same type of solutions could be obtained for other equations with singular points. The example above is convenient because it is a simple differential equation with explicit solutions.

But my question is this: is there a reason why solutions such as the one above are not considered? Is it just because it is too complicated to be included in textbooks? Perhaps because they do not extend to solutions when the independent variable is allowed to be complex? Or perhaps people shy away to solutions that are not smooth (i.e. not in $C^\infty$) when smooth solutions exist?

Gateau au fromage
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1 Answers1

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Most likely, the answer is pedagogical. It is a shame, though, that this idea isn't at least mentioned in standard undergraduate textbooks.

I will say, though, that you come across this sort of solution in quantum mechanics. For example, the finite square well problem requires you to solve the Schrödinger equation in three different regions, and then stitch them together to get continuity of the wave function and its derivative. You can see the solution method at the wiki for the Finite potential well.

So there are certainly applications for this kind of solution!

Incidentally, what's particularly intriguing to me about your question is that your proposed solution is continuous at the origin, and even its first few derivatives are continuous at the origin! So your problem there could certainly have physical significance.

Adrian Keister
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  • Thanks for the long and quick response. This question comes as a consequence of a problem which, like a Schrödinger equation, is an eigenvalue problem. Like the problem above, it involves a R-S point and I end up constructing eigenvectors like it is done above. The solutions are not explicit, but it is the same idea: solutions are constructed in a piecewise manner but they are continuous and do satisfy the equation everywhere, even at the singular point. I was thus looking for some advice or opinion of people who may have encountered problems like that. – Gateau au fromage Jul 12 '19 at 22:34