Page 141 - Advanced engineering mathematics

POWER SERIES SOLUTIONS
FROBENIUS SOLUTIONS
CHAPTER 4

Series Solutions

Sometimes we can solve an initial value problem explicitly. For example, the problem

y + 2y = 1; y(0) = 3
has the unique solution
1
y(x) = (1 + 5e −2x ).
2
This solution is in closed form, which means that it is a finite algebraic combination of elementary
functions (such as polynomials, exponentials, sines and cosines, and the like).
We may, however, encounter problems for which there is no closed form solution. For
example,
x
2
y + e y = x ; y(0) = 4

has the unique solution
x

ξ
2 e
y(x) = e −e x ξ e dξ + 4e −e x .
0
This solution (while explicit) has no elementary, closed form expression.
In such a case, we might try a numerical approximation. However, we may also be able to
write a series solution that contains useful information. In this chapter, we will deal with two
kinds of series solutions: power series (Section 4.1) and Frobenius series (Section 4.2).

4.1 Power Series Solutions

A function f is called analytic at x 0 if f (x) has a power series representation in some
interval (x 0 − h, x 0 + h) about x 0 . In this interval,
∞
n
f (x) = a n (x − x 0 ) ,
n=0
where the a n ’s are the Taylor coefficients of f (x) at x 0 :
1
a n = f (n) (x 0 ).
n!

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October 14, 2010 14:17 THM/NEIL Page-121 27410_04_ch04_p121-136