Page 304 - Schaum's Outline of Differential Equations

CHAP. 28] SERIES SOLUTIONS NEAR A REGULAR SINGULAR POINT 287

2
3
28.21. Find the indicial equation of x y" + xe*y' + (x - \)y = 0 if the solution is required near x = 0.
Here

and we have

2
from which/7 0 = 1 and q Q = —1. Using (_/) of Problem 28.20, we obtain the indicial equation as X - 1 = 0.

28.22. Solve Problem 28.9 by an alternative method.
2
The given differential equation, 3x y" — xy' + y = 0, is a special case of Ruler's equation

where bj(j=0, 1, ... , n) is a constant. Euler's equation can always be transformed into a linear differential equation
with constant coefficients by the change of variables

It follows from (2) and from the chain rule and the product rule of differentiation that

Substituting Eqs. (2), (3), and (4) into the given differential equation and simplifying, we obtain

Using the method of Chapter 9 we find that the solution of this last equation is y = c^ + c 2e <1/3)z . Then using (2)
1 113
and noting that e (1/3)z = (e ) , we have as before,

28.23. Solve the differential equation given in Problem 28.12 by an alternative method.
2
The given differential equation, x y" — xy' + y = 0, is a special case of Euler's equation, (_/) of Problem 28.22.
Using the transformations (2), (3), and (4) of Problem 28.22, we reduce the given equation to

z
The solution to this equation is (see Chapter 9) y = c^ + c 2ze . Then, using (2) of Problem 28.22, we have for the
solution of the original differential equation

as before.