Math 257 – Assignment 2
Due: Wednesday, January 19
1. For the following differential equations, verify that x0 = 0 is an ordinary point. Without
actually solving the equations, find then a lower bound for the radii of convergence of power
series solutions about x0 = 0:
a) (x2 + 2)y 00 + 2xy 0 = 0
b) (x − 1)y 00 + y 0 = 0
c) xy 00 + sin(x)y = 0
Hint: sin(x) = x −
x3 x5 x7
+
−
± ...
3!
5!
7!
2. For the following differential equations, verify that 0 is an ordinary point. Then find two
linearly independent series solution for each problem and determine their guaranteed radius
of convergence:
a) (x2 − 3)y 00 + 2xy 0 = 0
b)
y 00 − xy 0 − x2 y = 0
3. Consider the equation
x2 y 00 + x2 y 0 + y = 0.
Show that:
a) x0 = 0 is a singular point.
b) The equation has no non-trivial power series solution of the form y =
Hint: Show that an = 0 for all n ≥ 0.
P∞
n=0
an x n .
4. Find the first four non-zero terms to the series solution of the equation
(x2 − 4)y 00 + 3xy 0 + y = 0,
y(0) = 5,
y 0 (0) = 1.
5. Find a power series solution of the non-homogeneous equation y 00 − xy = 1 about the
ordinary point x0 = 0.