Problem Set 6 (Frobenius)

Method of Frobenius1. Solve the following equations using a series method.(i) 2x 2 y ′′ − xy ′ y 0 (ii) x 2 y ′′ xy ′ − y 0(iii) x 2 y ′′ 2xy ′ − 2y 0 (iv) xy ′′ 2y ′ 2xy 0(v) xy ′′ 3y ′ 4x 3 y 0 (vi) xy ′′ y ′ − xy 02. Use a series method to solve the Hermite equationIf H n x is the polynomial defined byy ′′ − 2xy ′ 2ny 0.H n x −1 n e x2show that H n satisfies the following relationsH n′ 2xH n − H n1d ndx n e−x2 ,′′ ′ ′H n 2xH n 2H n − H n1′H n1 2n 1H nHence show that H n x is a solution of the equation. Discuss the form of the secondsolution.3. Use a series method to solve the Laguerre equationIf L n x is the polynomial defined byshow that L n satisfies the relationsxy ′′ 1 − xy ′ ny 0.L n x e xxL n′xL n ′′ L n′dx dnn x2 e −x , nL n − nL n−1′ ′ nL n − nL n−1′ ′L n−1 L n L n−1 .Hence show that L n x is a solution of the equation. Discuss the form of the secondsolution.4. Show by the method of Frobenius that the two independent solutions of xy ′′ y 0aregiven by

1 x x −1.2 x2 x 31.2 2 .3 −……−1 m x m11.2 2 .3 2 …m 2 m 1 … 2 x 1 x 1 − x2 32 1 2 .2 2 …−1…m x m1 2 .2 2 …m − 1 2 m− 1 xlogx21 2 2 … 2m − 1 m15. If y satisfiesddxfxx − x 0. dydx gxy 0,where fx and gx are analytic in a neighbourhood of x 0 ,andfx 0 ≠ 0, prove that theroots of the indicial equation at x 0 always differ by an integer, but that the solution does notcontain a logarithmic term.