Explanation Given: The differential equation, 2 x y ″ + 5 ( 1 − 2 x ) y ′ − 5 y = 0 Approach: The point x = x 0 is called an ordinary point of the differential equation y ″ + p ( x ) y ′ + q ( x ) y = 0 If p and q are analytic at x = x 0 . The ratio test is, lim n → ∞ a n + 1 a n = l And l < 1 then series is convergent otherwise the series is divergent. The standard form of the Bessel’s equation of order p with regular singular point at x = 0 is given as, x 2 y ″ + x y ′ + ( x 2 − p 2 ) y = 0 The indicial equation is, r ( r − 1 ) + r − p 2 = 0 with roots r = ± p Where p is a nonzero constant. The general solution for the Bessel’s equation of order p is, y ( x ) = c 1 J p ( x ) + c 2 J − p ( x ) Where, J p = ∑ k = 0 ∞ ( − 1 ) k ( k + 1 ) ( k + p + 1 ) ( x 2 ) 2 k + p Consider, the equation a 2 ( x ) y ″ + a 1 ( x ) y ′ + a 0 ( x ) y = 0 … ( 1 ) Rewrite, the above equations as, y ″ + P ( x ) y ′ + Q ( x ) y = 0 ... ( 2 ) Here, P ( x ) = a 1 ( x ) a 2 ( x ) , Q ( x ) = a 0 ( x ) a 2 ( x ) , x ∈ I If P ( x ) and Q ( x ) are not analytic at x 0 ∈ I then x 0 is called a singular point of the differential equation ( 1 ) . When both of the functions, p ( x ) = ( x − x 0 ) P ( x ) and q ( x ) = ( x − x 0 ) 2 Q ( x ) are analytic at x = x 0 , the point x 0 is called a regular singular point. When x 0 = 0 , the equation ( 2 ) can be written as, x 2 y ″ + x p ( x ) y ′ + q ( x ) y = 0 … ( 3 ) Where p and q are assumed to be analytic at x = 0 . Assume that R denote the smaller of the radii of convergence of the power series expansion of p ( x ) and q ( x ) about x = 0 If r 1 and r 2 denote the roots of the indicial equation, r ( r − 1 ) + p ( 0 ) r + q ( 0 ) = 0 With r 1 ≥ r 2 . 1. If r 1 − r 2 ≠ integer , then y 1 ( x ) = x r 1 ∑ n = 0 ∞ a n x n , a 0 ≠ 0 y 2 ( x ) = x r 2 ∑ n = 0 ∞ b n x n , b 0 ≠ 0 2. If r 1 = r 2 = r , then y 1 ( x ) = x r ∑ n = 0 ∞ a n x n , a 0 ≠ 0 y 2 ( x ) = y 1 ( x ) ln x + x r ∑ n = 1 ∞ b n x n 3. If r 1 − r 2 = positive integer , then y 1 ( x ) = x r 1 ∑ n = 0 ∞ a n x n , a 0 ≠ 0 y 2 ( x ) = A y 1 ( x ) ln x + x r 2 ∑ n = 0 ∞ b n x n , b 0 ≠ 0 Calculation: Consider the differential equation, 2 x y ″ + 5 ( 1 − 2 x ) y ′ − 5 y = 0 … ( 4 ) Here, P ( x ) = 5 ( 1 − 2 x ) 2 x and Q ( x ) = 5 2 x Consider, p ( x ) = x P ( x ) = x ⋅ 5 ( 1 − 2 x ) 2 x = 5 ( 1 − 2 x ) 2 q ( x ) = x 2 Q ( x ) = 5 x 2 As p and q both are analytic at x = 0 , the point x = 0 is regular singular point. Rewrite the equation ( 4 ) . 2 x 2 y ″ + 5 x ( 1 − 2 x ) y ′ − 5 x y = 0 … ( 5 ) Compare the above equation with the standard form of Bessel’s equation. p ( x ) = 5 ( 1 − 2 x ) 2 and q ( x ) = 5 x Thus, p ( 0 ) = 5 2 and q ( 0 ) = 0 Substitute 5 2 for p ( 0 ) and 0 for q ( 0 ) in the indicial equation r ( r − 1 ) + p ( 0 ) r + q ( 0 ) = 0 . r ( r − 1 ) + 5 2 r = 0 r 2 − r + 5 r 2 = 0 r 2 + 3 r 2 = 0 Further solve, r ( r + 3 2 ) = 0 r = 0 , − 3 2 Thus, the roots are 0 , − 3 2 . As the roots are distinct, the solutions are, y 1 ( x ) = x 0 ∑ n = 0 ∞ a n x n , y 2 ( x ) = x − 3 / 2 ∑ n = 0 ∞ b n x n Thus, the general solution is, y ( x ) = x r ∑ n = 0 ∞ a n x n = ∑ n = 0 ∞ a n x n + r … ( 6 ) Differentiate equation ( 6 ) with respect to x . y ′ ( x ) = ∑ n = 0 ∞ ( n + r ) a n x n + r − 1 Again differentiate, y ″ ( x ) = ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) a n x n + r − 2 Substitute ∑ n = 0 ∞ ( n + r ) a n x n + r − 1 , ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) a n x n + r − 2 for y ′ and y ″ respectively in equation ( 5 ) . 2 x 2 ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) a n x n + r − 2 + 5 x ∑ n = 0 ∞ ( n + r ) a n x n + r − 1 + 5 x 2 ∑ n = 0 ∞ ( n + r ) a n x n + r − 1 − 5 x ∑ n = 0 ∞ a n x n + r = 0 2 ∑ n = 0 ∞ ( n + r ) ( n + r − 1 ) a n x n + r + 5 ∑ n = 0 ∞ ( n + r ) a n x n + r + 5 ∑ n = 0 ∞ ( n + r ) a n x n + r + 1 − 5 ∑ n = 0 ∞ a n x n + r − 1 = 0 Replace n by n − 1 and divide by x r in above equation

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Whether x=0 is an ordinary point or a regular singular point and the two linearly independent solutions to the given differential equation and state the maximum interval on which the solutions are valid.

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Whether x = 0 is an ordinary point or a regular singular point and the two linearly independent solutions to the given differential equation and state the maximum interval on which the solutions are valid.

Solution Summary: The author determines whether x=0 is an ordinary point or a regular singular point and states the maximum interval on which the solutions are valid.

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Whether x=0 is an ordinary point or a regular singular point and the two linearly independent solutions to the given differential equation and state the maximum interval on which the solutions are valid.

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