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Ted123
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HallsofIvy said:You should have y''+ py'+ qy= 0 and (x- x_0)^2q(x) finite. You have left out the "q"s. Also, you say that x_0 is a regular singular point if those are finite. In order that x_0 be a singular point, at least one of the two limits of p and q, as x goes to x_0 must not exist.
Yes, from what you have, x= 0 is a regular singular point for a and c, it not a regular singular point for d. But I would say that x= 0 is not even a singular point for b.
A regular singular point is a point in a differential equation where the solution can be expressed as a power series. It is called "regular" because the power series has a finite radius of convergence and does not have any singularities at the point.
A regular singular point is a point where the solution can be expressed as a power series, while an irregular singular point is a point where the solution cannot be expressed as a power series and may have infinite or non-integer powers.
Ordinary differential equations (ODEs) and partial differential equations (PDEs) can both have regular singular points. However, ODEs are more commonly studied in the context of regular singular points.
A regular singular point can be determined by examining the coefficients of the highest and second-highest order derivatives in the differential equation. If these coefficients are not constants, then the point is a regular singular point.
Regular singular points have applications in a variety of fields, including physics, engineering, and biology. They can be used to model physical systems and phenomena, such as the motion of a pendulum or the behavior of electrical circuits. They also have applications in the solution of differential equations in quantum mechanics and relativity.