4 d-B 4x4-12x2+3 16x3-24x 48x2-24 lI 20 60X iii. e3x+e-3x 3(e3x-e-3x)9( -4x+2 B(v)is an eigenfunction of A(i) (1-x2) dx b(v 24x-24x3-12x3+3x 36x3+27x 9(4x3-3x)(eigenvalue is-9) B(iii )is an eigenfunction of A(ii x)(eigenvalue is 9) B(ii )is an eigenfunction of A(iii) d9 4. B dB/dx d 2B/dx2 i. 4x4 - 12x2 + 3 16x3 - 24x 48x2 - 24 ii. 5x4 20x3 60x2 iii. e 3x + e-3x 3(e3x - e-3x) 9(e3x + e-3x) iv. x2 - 4x + 2 2x - 4 2 v. 4x3 - 3x 12x2 - 3 24x B(v.) is an eigenfunction of A(i.): (1-x2) d2 dx2 - x d dx B(v.) = (1-x2) (24x) - x (12x2 - 3) 24x - 24x3 - 12x3 + 3x -36x3 + 27x -9(4x3 -3x) (eigenvalue is -9) B(iii.) is an eigenfunction of A(ii.): d2 dx2 B(iii.) = 9(e3x + e-3x) (eigenvalue is 9) B(ii.) is an eigenfunction of A(iii.): x d dx B(ii.) = x (20x3)