Math 257 – Assignment 3
Due: Wednesday, January 26
1. For a constant λ, consider Hermite’s equation:
y 00 − 2xy 0 + λy = 0,
−∞ < x < ∞.
a) Find the first four nonzero terms in each of two linearly independent series solutions
centered at x0 = 0.
b) If λ = 2n is a non-negative even integer, then one or the other of the series solution
terminates and becomes a polynomial. Find these polynomials for λ = 8 and λ = 10.
c) The Hermite polynomial Hn (x) is defined as the polynomial solution with λ = 2n for
which the coefficient of xn is 2n . Find the polynomials H4 (x) and H5 (x).
2. For each of the following equations find all singular points and determine whether each one
is regular or irregular:
a) x3 y 00 + 4xy 0 = 0
c) xy 00 + y 0 +
b)
(x2 − 2x + 1)y 00 + y = 0
1+x
y=0
x
3. Determine the general solutions of the following Euler equations
a) x2 y 00 + 8xy 0 + 12y = 0,
x>0
b) (x − 2)2 y 00 + 5(x − 2)y 0 + 8y = 0,
x>2
4. The following equations have a regular singular point at x = 0 and their indicial equations
have two unequal roots that do not differ by an integer. For x > 0 find the first three non-zero
terms in each of two linearly independent series solutions:
a) 2xy 00 + y 0 + xy = 0
b) 2x2 y 00 + 3xy 0 + (2x2 − 1)y = 0
5. For the equation
x2 y 00 − 2xy 0 + (x + 2)y = 0,
x>0
verify that x = 0 is a regular singular point, find the indicial equation, the exponents at
the singularity, the recurrence relation, and the first three non-zero terms for the solution
corresponding to the larger root of the indicial equation.