Problems
In each of Problems 1 through 11:
a. Seek power series solutions of the given differential equation
about the given point $x_0$; find the recurrence relation that the
coefficients must satisfy.
b. Find the first four nonzero terms in each of two solutions $y_1$
and $y_2$ (unless the series terminates sooner).
c. By evaluating the Wronskian $W[y_1, y_2](x_0)$, show that $y_1$
and $y_2$ form a fundamental set of solutions.
d. If possible, find the general term in each solution.
1. $y'' - y = 0$, $x_0 = 0$
2. $y'' + 3y' = 0$, $x_0 = 0$
3. $y'' - xy' - y = 0$, $x_0 = 0$
4. $y'' - xy' - y = 0$, $x_0 = 1$
5. $y'' + k^2x^2y = 0$, $x_0 = 0$, $k$ a constant
6. $(1 - x)y'' + y = 0$, $x_0 = 0$
7. $y'' + xy' + 2y = 0$, $x_0 = 0$
8. $xy'' + y' + xy = 0$, $x_0 = 1$
9. $(3 - x^2)y'' - 3xy' - y = 0$, $x_0 = 0$
10. $2y'' + xy' + 3y = 0$, $x_0 = 0$
11. $2y'' + (x + 1)y' + 3y = 0$, $x_0 = 2$
