M334Lec22 - Math 334 Lecture#22 5.3 Power Series Solutions...

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Math 334 Lecture #22 § 5.3: Power Series Solutions, Part II Representation Principle. When can a general solution of y + p ( x ) y + q ( x ) y = 0 be represented by power series? If p ( x ) and q ( x ) are analytic at x 0 , then the general solution of the ODE is y = n =0 a n ( x - x 0 ) n = a 0 y 1 ( x ) + a 1 y 2 ( x ) , where y 1 and y 2 are linearly independent power series solutions each analytic at x 0 with a radius of convergence at least as large as the smallest of the radii of convergence of the power series expansions for p and q about x 0 . [Recall that a function is analytic about x 0 if it has a power series expansion about x 0 with a positive radius of convergence that equals the function on the interval of convergence.] The power series solutions y 1 and y 2 are solutions of the IVPS with initial conditions y 1 ( x 0 ) = 1 , y 1 ( x 0 ) = 0 , and y 2 ( x 0 ) = 0 , y 2 ( x 0 ) = 1 . The Wronskian of y 1 and y 2 at x = x 0 is 1, so that y 1 and y 2 are indeed linearly independent on the interval of convergence.
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