Another Cauchy-Euler ODE.
[tex]y=x^r[/tex]
[tex]\implies r(r-1)-13r+49=r^2-14r+49=(r-7)^2=0[/tex]
[tex]\implies r=7[/tex]
[tex]\implies y=C_1x^7+C_2x^7\ln x[/tex]
where [tex]y_1=x^7[/tex] and [tex]y_2=x^7\ln x[/tex].
To verify that these solutions are linearly independent, check the Wronskian:
[tex]W(y_1,y_2)=\begin{vmatrix}x^7&x^7\ln x\\7x^6&x^6(7\ln x+1)\end{vmatrix}=x^{13}[/tex]
and so the solutions are indeed independent.
