# Consider the differential equation 4x2y′′ − 8x2y′ + (4x2 + 1)y = 0 (a) Verify that x0 = 0 is a regular singular point of...

Consider the differential equation

4x2y′′ − 8x2y′ + (4x2 + 1)y = 0

(a) Verify that x0 = 0 is a regular singular point of the differential equation and then find one solution as a Frobenius series centered at x0 = 0. The indicial equation has a single root with multiplicity two. Therefore the differential equation has only one Frobenius series solution. Write your solution in terms of familiar elementary functions.

(b) Use Reduction of Order to find a second linearly independent solution and then form the general solution of the differential equation.

Note: In part (a), you will find one solution of the differential equation is y = √xex. You must show all of the work of finding the series in part (a), but if you cannot, proceed to reduction of order with the given function.

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