M285_6.2Ordinary_Points.pdf - Math 285 6.2 Solving about Ordinary Points Definition A function f is said to be analytic at a point a if it can be

M285_6.2Ordinary_Points.pdf - Math 285 6.2 Solving about...

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Math 285: 6.2 Solving about Ordinary Points Definition: A function f is said to be analytic at a point a if it can be represented by a power series in ( x ± a ) with either a positive ( R > 0) or an infinite radius of convergence ( R = ). Recall from Calc II : Functions like e x , cos x , sin x , ln(1 ² x ), tan ± 1 x , can be written as power series. Also polynomials of degree n can be written as power series of ( x ± a ) , with constants c n ² 1 , c n ² 2 , c n ² 3 ,... all equal to 0. Definition: A point x x 0 is said to be an ordinary point of the homogeneous linear second-order DE in standard form cc y ² P ( x ) c y ² Q ( x ) y 0 if both coefficients P ( x ) and Q ( x ) are analytic at x 0 . A point that is NOT an ordinary point of the DE is called a singular point . Differential Eq P ( x ) and Q ( x ) Singular Points cc y ² 6 c y ± 7 y 0 cc y ² cos x ³ ´ c y ² e x y 0 x cc y ² c y ² 3 x 2 y 0 ³ ´ 2 4 0 x y xy y cc c ± ² ± ³ ´ 2 4 0 x y xy y cc c ² ² ± NOTE: Given DE: a 2 ( x ) cc y ² a 1 ( x ) c y ² a 0 ( x ) y 0 , x x 0 is a singular point if a 2 ( x 0 ) 0 not undefined PCX
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