notes5-5 - EXAMPLE For the following differential equation...

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SECTION 5.5 SOLUTIONS NEAR REGULAR SINGULAR POINTS When x = 0 is a regular singular point for a differential equation P ( x ) y 00 + Q ( x ) y 0 + R ( x ) y = 0, we expect solutions to be a combination of Euler equation solutions and series solutions, that is, we expect to get a solution of the form y = x r X n =0 a n x n = X n =0 a n x n + r . We can always find at least one solution of this form when x = 0 is a regular singular point. Various complications arise that make the second linearly independent solution sometimes difficult to determine, and we won’t deal with these complications (See Section 5.6.).
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Unformatted text preview: EXAMPLE. For the following differential equation, determine the indicial equation, the recurrence relation, and the roots of the indicial equation. Find the series solution for x > corresponding to the larger root. If the roots are unequal and do not differ by an integer, find the series solution corresponding to the smaller root also. WE WILL BE SATISFIED WITH THE FIRST FOUR NONZERO TERMS OF THE SERIES. 2 x 2 y 00 + 3 xy + (2 x 2-1) y = 0 HOMEWORK: SECTION 5.5...
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