Consider differential equation xyxyy0 d Find the series solu

Consider differential equation: xy\'\'-xy\'-y=0

(d) Find the series solution y2(x) corresponding to the smaller root for x > 0. (Find the general formula for the coecients in the series.)

Solution

Rewrite the equation in the normal form

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

Comparing the coefficients, p(0) =0 and q(0) =0

So the indicial equation is

                        r(r-1)=0 , yielding r=0 ,1

The powerseries solution corresponding to the smaller root (r=0) is

y =   a[n]xⁿ

Substituting this in the given equation and equating the coefficients on both sides., we get

a[0]=0

a[2]-a[1]=0

...

a[n+1]= a[n]/n

or a[n+1] = 1/n!