Question $1$

a$)$ Apply the Leibnitz theorem to determine the $5^{th}$ derivative of the function $f(x)=x^{2}sin(3x)$

$(12$ marks$)$

b$)$ Apply the Leibnitz$-$Maclaurin method to determine the recurrence relation for $y^{(n+2)}$ in terms of $y^{(n+1)}$ and $y^{(n-1)}$ for the second order differential equation

$y^{\text{'}\text{'}}+y^{\text{'}}+2xy=0$

Note: do not go beyond finding the recurrence relation.

$(8$ marks$)$

Question $2$