Show that the functions $~f_1(x)=e^x,~ f_2(x)=xe^x,~ f_3(x)=x^2 e^x~$ are linear independent by using Wronskian .
2026-02-23 17:09:13.1771866553
wronskian for showing linear independent
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$$W(f_1,f_2,f_3)(x) =\begin{vmatrix} f_1 & f_2 & f_3 \\ f_1' & f_2' & f_3' \\ f_1'' & f_2'' & f_3'' \notag \end{vmatrix}= \begin{vmatrix} e^x & xe^x & x^2e^x \\ e^x & e^x(x+1) & e^x(2x+x^2) \\ e^x & e^x(x+2) & e^x(2+4x+x^2) \notag \end{vmatrix}$$ Can you take it from here?