SeriesOrdinaryPoint

SeriesOrdinaryPoint - Series Solutions Near an Ordinary...

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Series Solutions Near an Ordinary Point We are considering methods of solving second order linear equations when the coefficients are functions of the independent variable. We consider the second order linear homogeneous equation P(x) d 2 y dx 2 + Q(x) dy dx + R(x)y = 0 (1) since the procedure for the non-homogeneous equation is similar. Many problems in mathematical physics lead to equations of this form having polynomial coefficients; examples include the Bessel equation x 2 y’’ + xy’ + (x 2 – a 2 ) y = 0 where a is a constant, and the Legendre equation (1 – x 2 ) y’’ – 2xy’ + c(c + 1) y = 0 where c is a constant. For this reason, as well as to simplify the algebraic computations, we primarily consider the case in which P, Q, and R are polynomials. However, we will see that the method can be applied when P, Q, and R are analytic functions. For the time being, we assume P, Q, and R are polynomials and they do not have common factors. Suppose that we wish to solve equation (1) in a neighborhood of a point a. The solution of equation (1) in an interval containing point a is closely related with the behavior of P(x) in the interval. Definition : Given the equation P(x) d 2 y dx 2 + Q(x) dy dx + R(x)y = 0 the equation 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) is called the equivalent normalized form of equation(1). Definition : The point a is called an ordinary point of equation (1) if both of the functions p(x) and q(x) in the equivalent normalized form, are analytic functions at the point a. If either or both of these functions are not analytic at a, then the point a is a singular point of equation (1). Examples : 1) Given y’’ + xy’ + (x 2 + 2) = 0 since p(x) = x and q(x) = x 2 + 2 are polynomials, then they are analytic functions everywhere, then every real number is an ordinary point.
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2) Given (x 2 ! 1)y'' + xy' + 1 x y = 0 (x 2 – 1) the equivalent normalized equation is y'' + x x 2 ! 1 y' + 1 x x 2 ! 1 ( ) y = 0 where p(x) = x x 2 ! 1 and q(x) = 1 x x 2 ! 1 ( ) then the points 1, -1, and 0 are singular points of the equation, any other real number is an ordinary point of the equation. Theorem : If a is an ordinary point of the differential equation (1), then p(x) = Q(x)/P(x) and q(x) = R(x)/P(x) are analytic at a, and there are two nontrivial linearly independent power series solutions of equation (1) of the form c n (x ! a) n , n = 0 " # d n (x ! a) n n = 0 " # Furthermore, the radius of convergence for each of the series solutions is at least as large as the minimum of the radii of convergence of the series for p(x) and q(x). Example : 1) The point a = 0 is an ordinary point of the equation y’’ + (x 3 + 1) y’ – xy = 0 then, the differential equation has a solution in the form c n x n n = 0 ! " whose radius of
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SeriesOrdinaryPoint - Series Solutions Near an Ordinary...

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