Second Order Ode With Variable Coefficients

Part of Frobenius Series Solution:

Where to avoid confusion, I have used a different dummy variables m on the left from the n on the right. Since this equation has to be true for a range of value of x around the origin, it must be true seperately for each power of x. We must therefore compare the coefficient of different power of x on the two sides of the equations. To facilitate the comparision, let m = n 1 on the left hand side.

Since we now have power x^{n k} explicitly on both sides of the equation, it is simple to deduce that ( n k 1) a_{n 1} = αa_n

This is very simple example of a recurrence relation, which allows us to evaluate all the heigher coefficients from the first one. It does however contain the unknown index k. How is its value fixed? The lowest power of x on the left hand side of the above equations is a₀kx^k-1and there is nothing like this on the right hand side bacause the sum there starts with n = 0. The term must therefore be made to cancel.

Since a₀ ≠ 0, this can only happen if k = 0. Above equation is a very simple example of what is called an indicial equation; it fixes the index k. With k = 0, the recurrence relation becomes

Now, let a₀ = A. Then