Consider the Chebyshev differential equation (1 – x²)y" – xy' + X?y = 0, where A is a constant. Find the power series solutions of this differential equation about xo = 0. For what values of the constant A these solutions reduce to polynomials, known as Chebyshev polynomials. Compute first four of the Chebyshev polynomials.

If y=Σanxn(n from zero to infinity) is the solution to the following differential equation, solve the equations using power series and find the sequence an. y''+x2y=0

Solve the differential equation using either the method of undetermined coefficients or Laplace transforms. y'''+y"+y'+y=e-t+4t

The point x = O is a regular singular point of the given differential equation. Find the recursive relation for the series solution of the DE below. Show the substitution and all the steps to obtain the recursive relation. Do not solve the equation for y=y(x) xy" + 4y' - xy = 0,