Motivation: Finite difference methods are an important method for approximating solutions of differential equations. We have seen an example in Lecture 05, pp. 13-15, where the solution of a boundary value problem was approximated using finite differences. A similar method can be used for initial value problems, as in the first part below. Finite difference methods will be an important topic again in Lecture 11 when we discuss solving the Black Scholes PDE in pricing financial options. The following example is chosen for practice. An approximation to the solution is implemented in Matlab ( besselj). This gives us an easy way to check the accuracy of our solution. Consider the initial value problem x2u(x)+xu(x)+(x22)u(x)=0,0