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Solution about singular points. Definition. Theorem. Example 1. Use Frobenius method to solve given differential equation about a regular singular point x = 0,3 x y ' ' + y ' y = 0 . Solution. We will try to find a solution of the form Then

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So
Then

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And y = C 1 y 1 ( x ) + C 2 y 2 ( x ) is solution given differential equation. Bessel’s and Legendre’s equations. Definition. The differential equation x 2 y ' ' + x y ' + ( x 2 ν 2 ) y = 0 is called Bessel’s equation of order ν . The differential equation 1 x ( ¿¿ 2 ) y ' ' 2 x y ' + n ( n + 1 ) y = 0 ¿ is called Legendre’s equation of order n . The general solution of Bessel’s equation is y = C 1 J ν ( x ) + C 2 J ν

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Unformatted text preview: ν ( x ) , where J ν ( x ) = ∑ n = ∞ (− 1 ) n n! Г ( 1 + ν + n ) ( x 2 ) 2 n + ν and J − ν ( x ) = ∑ n = ∞ (− 1 ) n n! Г ( 1 − ν + n ) ( x 2 ) 2 n − ν are Bessel’s functions, Г ( x )= ∫ ∞ t x − 1 e − t dt . Example. 1. The solution of the equation x 2 y ' ' + x y ' + ( x 2 − 1 4 ) y = is y = C 1 J 1 2 ( x ) + C 2 J − 1 2 ( x ) , because ν 2 = 1 4 and ν = 1 2 . 2. The general solution of Legendre’s equation is y = C 1 y 1 ( x ) + C 2 y 2 ( x ) , where...
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