Question

Chebyshev's equation of order p:

a) Solve (1-x² )y'' - xy' + p²y = 0 using power series methods at x₀ = 0.
b) For what p is there a polynomial solution?

Answer 1

To solve (1-x² )y'' - xy' + p²y = 0 using power series methods at x₀ = 0, assume a power series solution and solve for its coefficients. Determine the values of p for which there is a polynomial solution.

Step-by-step explanation:

To solve the differential equation (1-x² )y'' - xy' + p²y = 0 using power series methods at x₀ = 0, we can assume a power series solution of the form y = ∑n=0∞anxn. Substituting this into the equation and equating coefficients of like powers of x, we can solve for the coefficients an. By finding the values of p for which the power series solution is a polynomial, we can determine the values of p for which there is a polynomial solution.