(a) Define operators : $\Delta, \nabla, E$. Also evaluate $\Delta(e^x…

Question

(a) Define operators : $\Delta, \nabla, E$. Also evaluate $\Delta(e^x \cos 2x)$ and $\nabla(x^2 \sin x)$.

(b) The following table gives the values of $f(x) = \tan x$ for $x$ lying within $0.10$ and $0.30$ :

| $x$ | $f(x)$ |

|---|---|

| 0.10 | 0.1003 |

| 0.15 | 0.1511 |

| 0.20 | 0.2027 |

| 0.25 | 0.2553 |

| 0.30 | 0.3093 |

Find the value of $\tan(0.26)$.

(c) Solve the difference equation : $y_{n+2} - 2y_{n+1} + 4y_n = n^2 . 6^n$.