RegularSingularPoints

RegularSingularPoints - Regular Singular Points We now...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Regular Singular Points We now investigate the solution of the homogeneous second-order linear equation P(x) d 2 y dx 2 + Q(x) dy dx + R(x)y = 0 (1) near a singular point. Recall that if the functions P(x), Q(x), and R(x) are polynomials having no common factors, then the singular points of the equation (1) are simply those points where P(x) = 0. Example : 1) The point x = 0 is the only singular point of the Bessel equation of order n, x 2 y’’ + xy’ +(x 2 – n 2 )y = 0 2) The points x =1 and x = -1are the two singular points of the Legendre equation (1 –x 2 )y’’ -2xy’ + n(n+1)y = 0 It turns out that some important features of the solutions of such equations are largely determined by the behavior near their singular points. Remark : We will restrict our attention to the case in which x = 0 is a singular point of the equation. A differential equation having x = a as a singular point is easily transform by the substitution t = x – c into one having the corresponding singular point at 0. Types of Singular Points A differential equation having a singular point at 0, in general, will not have power series solution of the form y(x) = c n x n n = 0 ! " , so the straightforward method of the previous section fails in this case. To investigate the form that the solution of such equation might take, we assume that equation (1) has analytic coefficient functions and we rewrite it in standard form, d 2 y dx 2 + Q(x) P(x) dy dx + R(x) P(x) y = 0 or d 2 y dx 2 + p(x) dy dx + q(x)y = 0 where p(x) = Q(x) P(x) and q(x) = R(x) P(x) Recall that x = 0 is an ordinary point (as opposed to a singular point) of equation if both p(x) and q(x) are analytic functions at x = 0, this means that they both have a power series expansion around 0. It can be proved that each of the functions p(x) and q(x) either is analytic or approach ± as x approaches 0. Consequently, x = 0 is a singular point of the equation provided that either p(x) or q(x) (or both) approaches ± as x approaches 0. Definition : Given the normalized equation
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
d 2 y dx 2 + Q(x) P(x) dy dx + R(x) P(x) y = 0 or d 2 y dx 2 + p(x) dy dx + q(x)y = 0 where p(x) = Q(x) P(x) and q(x) = R(x) P(x) let x 0 be a singular point. If the functions A(x) = (x – x 0 ) p(x), and B(x) = (x – x 0 ) 2 q(x) are both analytic at x 0 , then x 0 is called a regular singular point of the equation. If either (or both) of the functions A(x) or B(x) is not analytic at x 0 , then x 0 is called an irregular singular point. Example : 1) Given (1 + x)y’’ + 2xy’ – 3y = 0, the normalized equation is y'' + 2x 1 + x y' ! 3 1 + x y = 0 Then, x = -1 is a singular point of the equation since p(x) and q(x) are not analytic at -1. Since A(x) = (x +1) p(x) = 2x, and B(x) = (x +1) 2 q(x) = 3(x + 1) are both analytic functions at x = -1, then x = -1 is a regular singular point. 2) Given x
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 16

RegularSingularPoints - Regular Singular Points We now...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online