Given [tex]y_1=e^{6x}[/tex], assume a second solution of the form [tex]y_2=vy_1[/tex], with derivatives
[tex]{y_2}'=v'y_1+v{y_1}'[/tex]
[tex]{y_2}''=v''y_1+2v'{y_1}'+v{y_1}''[/tex]
With [tex]y_1=e^{6x}[/tex], you have [tex]{y_1}'=6e^{6x}[/tex] and [tex]{y_1}''=36e^{6x}[/tex].
Substitute these into the ODE and you get
[tex](e^{6x}v''+12e^{6x}v'+36e^{6x}v)-12(e^{6x}v'+6e^{6x}v)+36e^{6x}v=0[/tex]
[tex]v''+12v'=0[/tex]
Now substitute [tex]w=v'[/tex], so that [tex]w'=v''[/tex] and you have a linear first-order ODE:
[tex]w'+12w=0\implies e^{12x}w'+12e^{12x}w=(e^{12x}w)'=0\implies e^{12x}w=C[/tex]
[tex]\implies w=v'=Ce^{-12x}[/tex]
[tex]\implies v=C_1e^{-12x}+C_2[/tex]
[tex]\implies y_2=(C_1e^{-12x}+C_2)e^{6x}=C_1e^{-6x}+C_2e^{6x}[/tex]
But [tex]y_1=e^{6x}[/tex] is already accounted for, so the second fundamental solution to the ODE is [tex]y_2=e^{-6x}[/tex].
