Consider the Chebyshev differential equation(1−x2)y″−xy′+λ2y=0where λ is a constant. Find the power series solutions of this differential equation about x0=0. For what values of the constant λ these solutions reduce to polynomials, known as Chebyshev polynomials. Compute first four of the Chebyshev polynomials.

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bahaistag · Accepted Answer

The objective is to find power series solution to Chebyshev differential equation(1−x2)y″−xy′+λ2y=0about the point x0=0 and λ is a constant.Also, we have to find the values of λ such that solution reduces to Chebyshev polynomials and we have to estimate first four Chebyshev polynomials.Compare the equation (1−x2)y″−xy′+λ2y=0 with P(x)y″+Q(x)y′+y=0.Here, P(x)=1−x2 and Q(x)=−xBoth the functions P(x)=1−x2 and Q(x)=−x are analytic at x0=0.Now, we look for solution of the form y(x)=∑n=0∞anxnSubstitute the solution in the original equation (1−x2)y″−xy′+λ2y=0y′(x)=∑n=0∞nanxn−1y″(x)=∑n=0∞n(n−1)anxn−2Therefore we have,∑n=2∞n(n−1)anxn−2−∑n=2∞n(n−1)anxn−∑n=1∞nanxn+∑n=0∞λ2anxn=0Now replace n by n+2 in first term and simplify∑n=0∞(n+2)(n+1)an+2xn−∑n=2∞n(n−1)anxn−∑n=1∞nanxn+∑n=0∞λ2anxn=0

Consider the Chebyshev differential equation. (1-x^2)y''-xy'+\lambda^2y=0. where \lambda is a constant

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