Section 5.3 Frobenius Method

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eqn=x (x-1) y''[x]+(3x-1) y'[x]+ y[x]==0

sol=DSolve[eqn,y,x]

sol /. {C[1] -> 1, C[2] -> 0}

y1[x_]:=y[x] /.sol /. {C[1] -> 1, C[2] -> 0}

y1[x]

y2[x_]:=y[x]/.sol/. {C[1]->0, C[2]-> 1}

y2[x]

Plot[{y1[x],y2[x]},{x,-2,6},PlotLegends->"Expressions"]

eqn=x (x-1) y''[x]-x y'[x]+ y[x]==0

sol=DSolve[eqn,y,x]

y1[x_]:= y[x] /.sol /. {C[1] -> 1, C[2] -> 0}

y1[x]

y2[x_]:= y[x] /.sol /. {C[1] -> 0, C[2] -> -1}

y2[x]

Plot[{y1[x],y2[x]},{x,0,6},PlotLegends->"Expressions"]