This Appendix addresses the question of solving a linear ordinary second-order homogeneous differential equation

We do not go in to the conditions on the functions P(x) and Q(x) required for the equation above to admit a solution but assume that the functions possess the desired properties. The explicit solution will depend, of course, on the functional forms of P(x) and Q(x). However, before finding explicit solutions, we list below some properties of the solutions of (B.1) which are independent of the functional forms of P(x) and Q(x).

• 1. If y1(x) and y2(x) are two solutions of (B.1) then on substituting in it y(x) = ay1(x) + βy2(x), where a and b are complex numbers, it may be seen that y(x) also satisfies that equation.

• 2. Since, as shown above, any linear combination of the solutions of (B.1) is also a solution, it is sufficient to find all its linearly independent solutions. Any other solution can then be expressed as a linear combination of those linearly independent ones. To ascertain whether the solutions y1(x), y2(x) are linearly independent, assume that there exist constants A and B such that

The functions y1(x), y2(x) are linearly independent if the equation above is solved only by A = B = 0. The constants A, B are determined by forming second equation by differentiating (B.2) and solving the equation so obtained simultaneously with (B.2) by writing them as