Exam 2

All non-human resources are allowed.

Consider the Chebyshev differential equation

Find an integrating factor μ  to write the equation in standard Sturm Louiville form.

Plug the power series  into the original ODE and match powers of x to find a recurrence relationship for (Subscript[a, n]).

Numerically compute the first few terms of the solution with (Subscript[a, 0])=1 and (Subscript[a, 1])=0

Numerically compute the first few terms of the solution with (Subscript[a, 0])=0 and (Subscript[a, 1])=1

FYI: The Mathematica command for the standard bounded solution on [-1,1] is  ChebyshevT[p,x].  The equivalent matlab command is chebyshevT.

Make a table of the formulas for the first kind function (Ts) for n=0,1,2,3,4,5. Plot and label them together.

The Ts are orthogonal on -1<x<1 with respect to the weight (1/(Sqrt[1-(x^2)])).

Verify this orthogonality for several pairs of first kind functions.

Assume that a function f[x]=(∑_(i=0)^∞)(Subscript[b, n])(Subscript[T, n])(x)  on -1<x<1.

Explain how you use orthogonality to compute the coefficients (Subscript[b, n]).

Approximate Sin[x-12Cos[x]] on (-1,1) as a sum of (Subscript[T, n])(x) use 4, 8 and 12 terms.

Illustrate the convergence by plotting and labelling a few different approximations (different numbers of terms) and function on the same axes.  Comment on any interesting features.

Illustrate the convergence by plotting and labelling the relative error for a few different approximations on the same axes.  Comment on any interesting features.