Answers: 1
be an auxiliary solution of equation (i).
![m^3e^{mx}-m^2e^{mx}+2e^{mx}=0\\\\\Rightarrow (m^3-m^2+2)e^{mx}=0\\\\\Rightarrow m^3-m^2+2=0~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }e^{mx}\neq0]\\\\\Rightarrow m^2(m+1)-2m(m+1)+2(m+1)=0\\\\\Rightarrow (m+1)(m^2-2m+2)=0\\\\\Rightarrow m+1=0~~~~~\Rightarrow m=-1](/tpl/images/0083/3615/b5759.png)
![m^2-2m+2=0\\\\\Rightarrow (m^2-2m+1)+1=0\\\\\Rightarrow (m-1)^2=-1\\\\\Rightarrow m-1=\pm\sqrt{-1}\\\\\Rightarrow m=1\pm i~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{where }i^2=-1]](/tpl/images/0083/3615/ce997.png)
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