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**slapmaxwell1** Consider $\displaystyle L(y)=e^{2x}$

the Wronskian is W = $\displaystyle \begin{vmatrix}

e^{x} & e^{2x}&e^{3x} \\

e^{x}& 2e^{2x} & 3e^{2x}\\

e^{x}&4e^{2x} & 9e^{3x}

\end{vmatrix}=2e^{6x}.$

ok so now I am supposed to write an annihilator for L(y)=0,

then write the complementary solution $\displaystyle y_{c}$,

write an annihilator for $\displaystyle L(y)=e^{2x}$

rewrite $\displaystyle L(y)=e^{2x}$ in its expanded $\displaystyle (y,y',...)$

notation.

so the annihilator is $\displaystyle (D-1)(D-2)(D-3)$

$\displaystyle y_{c}=C_{1}e^{x}+C_{2}e^{2x}+C_{3}e^{3x}$

the last two parts of the problem im not sure how to answer, and to be honest im not sure if the work i have done up to this point is correct. any help would be appreciated.