118 n0 on the other hand x2 y an xnr2 n0 am2

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Unformatted text preview: 1.3.3. We want to find generalized power series solutions to the differential equation d2 y dy 3x 2 + +y =0 dx dx 21 by the method of Frobenius. Our procedure is to find solutions of the form ∞ ∞ y = xr an xn = n=0 an xn+r , n=0 where r and the an ’s are constants. a. Determine the indicial equation and the recursion formula. b. Find two linearly independent generalized power series solutions. 1.3.4. To find generalized power series solutions to the differential equation 2x d2 y dy + + xy = 0 dx2 dx by the method of Frobenius, we assume the solution has the form ∞ an xn+r , y= n=0 where r and the an ’s are constants. a. Determine the indicial equation and the recursion formula. b. Find two linearly independent generalized power series solutions. 1.4 Bessel’s differential equation Our next goal is to apply the Frobenius method to Bessel’s equation , x d dx x dy dx + (x2 − p2 )y = 0, (1.17) an equation which is needed to analyze the vibrations of a circular drum, as we mentioned before. Here p is a parameter, which will be a nonnegative integer in the vibrating drum p...
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