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Section 6.2 April 8, 2011
MTH 225 Differential Equations
Notes 6.2 Solutions about Singular Points (Infinite Series Solutions)
Consider the two DEs: P. Seeburger
and
What do they have in common?
Which can be solved using a power series centered at x = 0?
Quick Review:
A function f is analytic at a point a if it can be represented by a power series centered at a with R > 0.
Definition: Suppose the linear 2nd­order DE
is put into standard form by dividing through by the leading coefficient a2(x): Then: A point x0 is said to be an ordinary point of the DE if both P(x) and Q(x) from the standard form above are analytic at x0. A point that is not an ordinary point is called a singular point of the equation.
A New Definition:
A singular point x0 of a linear DE of the form is said to be a ________________________ singular point of the DE if the functions are both analytic at x0. and Otherwise, it is called an ____________________________ singular point of the equation.
Examples:
(a Cauchy­Euler equation)
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Section 6.2 April 8, 2011
If x − x0 appears at most to the first power in the denominator of P(x) and at most to the second power in the denominator of Q(x), then x = x0 is a regular singular point.
Frobenius’ Theorem
Theorem 6.2.1
If x = x0 is a regular singular point of the DE , then there exists ______________________________________ solution of the form:
Where the number r is a number to be determined. The series will converge at least on some interval:
0 < x − x0 < R.
Finding an Infinite Series Solution using Frobenius’ Theorem:
To find an infinite series solution of this form, we will assume that we can write the solution in the form As in Section 6.1, we find y′ and y′′ and substitute these series into the DE. Next we will combine the series into one expanded about xn. Then although we will again use a variation of the method of undetermined coefficients to determine all of the coefficients cn (usually in terms of c0 and c1),
we will also need to determine the _______________________________. Sometimes we will only find one solution this way. In this case, it is possible to use the reduction of order formula from Section 4.2 to find a second solution (although it is more difficult to use this formula on a series solution, as you might guess).
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Section 6.2 April 8, 2011
Ex. 1: Solve 3
Section 6.2 April 8, 2011
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Section 6.2 April 8, 2011
Ex. 2: Solve For 1st part of solution, see the PDF notes on Website (or on handout)
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Three Cases: Let us assume that r1 and r2 are the indicial roots of a DE for a regular singular point x = 0,
such that _______________________.
Case 1: If the difference r1 − r2 is not a positive integer, then there exist two linearly independent solutions of the DE in the form:
Case 2: If the difference r1 − r2 is a positive integer, then there exist two linearly independent solutions of the DE in the form:
Where C is a constant that could be zero.
Case 3: If r1 = r2, then there exist two linearly independent solutions of the DE in the form:
Same as Case 2 results, but with C = _____.
In Case 2, we may or may not obtain 2 linearly independent solutions from the method of Frobenius.
In Case 3, the method of Frobenius will never produce a second solution.
When only one solution is obtained through the method of Frobenius, we can use the reduction of order formula to obtain the other solution.
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