Consider the following differential equation,

xy9y+2xy=0.

Note:For each part below you must give your answers in terms of fractions (as appropriate), not decimals.

(a) The above differential equation has a singular point atx=0

. If the singular point atx=0

is aregularsingular point, then a power series for the solutiony(x)

can be found using the Frobenius method. Show thatx=0

is a regular singular point by calculating:

xp(x)= ?

x^(2)q(x)= ?

Since both of these functions are analytic atx=0

the singular point is regular.

(b) Enter the indicial equation, in terms ofr

, by filling in the blank below.

? =0

(c) Enter the roots to the indicial equation below. You must enter the roots in the order of smallest to largest, separated by a comma ?

(d) You must now calculate the solution for the largest of the two indicial roots.

First, enter the corresponding recurrence relation below, as an equation.

Note 1:You must include an equals sign.

Note 2:You must use the symbolmas your index.

Note 3:a

m

is entered asa(m),a

m+1

asa(m+1), etc. ?

(e) Hence enter the first three non-zero terms of the solution corresponding to the largest indicial root.

Note:The syntax fora0

anda1

isa0anda1,respectively.

y= ?