Warm–up Problems Find two power series solutions of the DE y + xy + y = 0 about the ordinary point x = 0.

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Presentation transcript:

Warm–up Problems Find two power series solutions of the DE y + xy + y = 0 about the ordinary point x = 0.

Solutions About Singular Points x0 is a singular point of y + P(x)y + Q(x)y = 0 if P(x) or Q(x) are not analytic at x0 Usually this means we’re dividing by 0. Def. A singular point x0 is regular if x0 is a root at most once in the denominator of P(x) and at most twice in the denominator of Q(x).

Ex. (x2 – 4)2y + 3(x – 2)y + 5y = 0

Thm. Frobenius’ Theorem If x0 is a regular singular point, then there exists at least one solution of the form where r is some constant. Using this is called the Method of Frobenius The solution is an infinite series, but it may not be a power series.

Ex. 3xy + y – y = 0

 r(3r – 2) = 0 is called the indical equation r = 0 and r = 2/3 are called the indical roots Note that xP(x) and x2Q(x) are analytic, so xP(x) = a0 + a1x + … x2Q(x) = b0 + b1x + …  The indical equation is r(r – 1) + a0r + b0 = 0

Ex. Find the indical roots of 2xy + (1 + x)y + y = 0

Ex. xy + y = 0

The values of r determine our solutions: Case 1: r1 – r2 is not an integer Frobenius will give us two solutions

Case 2: r1 – r2 is an integer or r1 = r2 We may still get two solutions from Frobenius If not, we use reduction of order to find the second solution

Ex. Find the second solution to xy + y = 0