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Partial Differentials Notes_Part_8

# Partial Differentials Notes_Part_8 - Remark The definition...

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Remark : The definition of a singular point assumes that the other coefficients do not both vanish there, i.e., either q ( x 0 ) negationslash = 0 or r ( x 0 ) negationslash = 0. If all three functions happen to vanish at x 0 , we can cancel any common factor ( x x 0 ) k , and hence, without loss of generality, can assume at least one of the coefficients is nonzero at x 0 . Proofs of the basic existence theorem for differential equations at regular points can be found in [ 66 , 72 ]. Theorem 12.5. Let x 0 be a regular point for the second order homogeneous linear ordinary differential equation (12.69) . Then there exists a unique, analytic solution u ( x ) to the initial value problem u ( x 0 ) = a, u ( x 0 ) = b. (12 . 71) The radius of convergence of the power series for u ( x ) is at least as large as the distance from the regular point x 0 to the nearest singular point of the differential equation in the complex plane. Thus, every solution to an analytic differential equation at a regular point x 0 can be expanded in an convergent power series u ( x ) = u 0 + u 1 ( x x 0 ) + u 2 ( x x 0 ) 2 + · · · = summationdisplay n =0 u n ( x x 0 ) n . (12 . 72) Since the power series necessarily coincides with the Taylor series for u ( x ), its coefficients u n = u ( n ) ( x 0 ) n ! are multiples of the derivatives of the function at the point x 0 . In particular, the first two coefficients u 0 = u ( x 0 ) = a, u 1 = u ( x 0 ) = b. (12 . 73) are fixed by the initial conditions. Once the initial conditions have been specified, the remaining coefficients must be uniquely prescribed thanks to the uniqueness of solutions to initial value problems.

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