The Frobenius Method

This method is used to obtain series solutions of differential equations of thetype f(x)y′′ + g(x)y′ + h(x)y = 0.

1. Assume a solution of the form y =∞∑

n=0

anxn+c, a0 6= 0, where c is a constant

to be determined. This is called a Frobenius series.

2. Substitute this series in the differential equation.

3. Equate the coefficients of the lowest term (often xc) to zero to get a quadraticequation for c, called the indicial equation. Hence find two values for c: c1and c2.

4. Equate the coefficients of the next lowest term(s) to zero. This step may notbe necessary in certain cases, but may need to be repeated in other cases.

5. Equate the coefficients of the general term to zero to obtain a recurrence rela-tion. Hence an expression y(x, c) for the solution of the differential equationis obtained.

6. The following cases may arise:

(a) if c1 − c2 is not an integer, substitute c = c1 and c = c2 in y(x, c) toobtain two linearly independent solutions;

(b) if c1 = c2, then substitute c = c1 in y(x, c) and in∂

∂cy(x, c) to obtain

two linearly independent solutions;

(c) if c1 − c2 is a non-zero integer, then

i. if indeterminate coefficients occur for one value of c, then substitutethis value in y(x, c) to obtain two linearly independent solutions;

ii. if infinite coefficients arise for c = c2, substitute c = c1 in y(x, c)

and c = c2 in∂

∂c[(c− c2)y(x, c)] to obtain two linearly independent

solutions.

JLB MAT2003