Use the method of Frobenius to obtain linearly independent series solutions abou

Question

Use the method of Frobenius to obtain linearly independent series solutions about x=0.

9x^(2)y+9xy+(x^(2)4)y=0.

Use an initial index of k=2 to develop the recurrence relation.

The indicial roots are(in ascending order) r1= , r2= .

Corresponding to the larger indicial root, the recurrence relation of the solution is given by
ck=    ×c(k2). The initial index is k= ,

The solution is
y1=c0(1x(1)+2x(2)+3x(3)+4x(4)+5x(5)+…)
where 1= , 1= , 2= , 2= , 3= , 3= , 4= , 4= , 5= , and 5= .
Note: the i's and i's are constants.

Explanation / Answer

Dividing by 9x2 , write the equation in the standard form

                                         y" + p(x)/x2 y' + q(x)y =0.

Thus p(x) =1 and q(x) = (4-x2 )/9

The indicial equation is

                                      r (r-1) + p(0)r + q(0) = 0

yieds                              r(r-1) + r -4/9 =0

So the roots are -2/3 and 2/3.in the ascending order.

To obtain the solution corresponding to r =2/3

set y(x) = xr (c[0] + c[1]x + c[2]x2 +.......) in the differential equation , and compare the coefficients of both sides to obtain the recurrence relation

                       c[k] = - c[k-2]/{(r+k)(r+k-1)   if k is even

otherwise it is 0.

Plugging the values in this we obtain the reqired values.

For example, c[2] = -c[0]/ 2   and so on.

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