DO5

Determine $D y m$ , in terms of $\left [ \lambda^j _k y \right ] y^l$

$D = \frac{d^2}{dx^2} + \frac{d}{dx} + 1 = \lambda_2 + \lambda_1 +\lambda_0$

$λ 0 y m = y m$

$λ 1 y m = m y m - 1 λ 1 y$

$\lambda_2 y^m = m(m-1)y^{m-2} \lambda^2_1 y+ my^{m-1} \lambda_2 y$

Hence $Dy^m = y^m + \left [ m(\lambda_1 y + \lambda_2 y) \right ] y^{m-1} + \left [ m(m-1)\lambda_1^2 y \right ] y^{m-2}$