Find the indicial roots of the differential equation 2xy''+5y'+xy=0 around x=0.

Question

Find the indicial roots of the differential equation  around .

Answer 1

Rewrite: y"+5y'/2x+xy/2x=0=y"+5y'/2x+y/2=0.

Let P(x)=1/2x and Q=½, then the indicial equation is r(r-1)+p₀r+q₀=0, where p(x)=xP(x)=½, q(x)=x²Q(x)=x²/2,

so p₀=p(0)=½, q₀=q(0)=0.

Therefore the indicial equation becomes:

r(r-1)+r/2=0=r²-r+r/2=r²-r/2=r(r-½), from which r=0 or ½.