91Ó°ÊÓ

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$4 x^{2} y^{\prime \prime}+4 x(1+2 x) y^{\prime}+(4 x-1) y=0$$

Determine two linearly independent solutions to the given differential equation on \((0, \infty)\) $$4 x^{2} y^{\prime \prime}-(3+4 x) y=0$$

Determine a Frobenius series solution to the given differential equation and use the reduction of order technique to find a second linearly independent solution on \((0, \infty)\) $$x y^{\prime \prime}-x y^{\prime}+y=0$$

Determine a Frobenius series solution to the given differential equation and use the reduction of order technique to find a second linearly independent solution on \((0, \infty)\) $$x^{2} y^{\prime \prime}+x(4+x) y^{\prime}+(2+x) y=0$$

Consider the Laguerre differential equation $$x^{2} y^{\prime \prime}+x(1-x) y^{\prime}+N x y=0$$ where \(N\) is a constant. Show that in the case when \(N\) is a positive integer, Equation \((11.5 .40)\) has a solution that is a polynomial of degree \(N,\) and find it. When properly normalized, these solutions are called the Laguerre polynomials.

Consider the differential equation $$x^{2} y^{\prime \prime}+x(1+2 N+x) y^{\prime}+N^{2} y=0, \quad(11.5 .41)$$ where \(N\) is a positive integer. (a) Show that there is only one Frobenius series solution and that it terminates after \(N+1\) terms. Find this solution. (b) Show that the change of variables \(Y=x^{N} y\) transforms Equation \((11.5 .41)\) into the Laguerre differential equation (11.5.40).