Math 334
Assignment 7
Due: 12 Noon on Thursday, November 9, 2006.
1. Chebyshev’s differential equation is
(1 −x2)y′′ −xy′+α2y= 0,
where αis a constant.
(a) Find two linearly independent power series solutions valid for |x|<1.
(b) Show that if α=nis a non–negative integer, then there is a polynomial solution of degree n.
These polynomials, when properly normalised, are called Chebyshev polynomials.
(c) Find a polynomial solution for each of the cases α=n= 0,1,2,3.
2. For each of the following differential equations, locate and classify its singular p oints.
(a) x3(x−1)y′′ −2(x−1)y′+ 3xy = 0.
(b) x2(x2−1)2y′′ −x(x−1)y′+ 2y= 0.
3. Find the indicial equation and its roots for each of the following differential equations:
(a) x3y′′ + (cos 2x−1)y′+ 2xy = 0;
(b) 4x2y′′ + (2x4−5x)y′+ (3x2+ 2)y= 0.
4. For each of the following equations, verify that the origin is a regular singular point and calculate two
independent Frobenius series solutions:
(a) 4xy′′ + 2y′+y= 0;
(b) 2xy′′ + (3 −x)y′−y= 0.
5. Consider the differential equation
x3y′′ +xy′−y= 0.
(a) Show that x= 0 is an irregular singular point.
(b) Use the fact that y1(x) = xis a solution to find a second independent solution y2(x).
(c) Show that the solution y2(x) found in part (b) cannot be expressed as a Frobenius series.