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Solving a Third Order Constant Coefficient Linear Homogeneous Differential Equation

Consider the linear differential constant coefficient linear equation 
\[\frac{d^3 y}{dx^3}-6 \frac{d^2y}{dx^2}+11\frac{dy}{dx}-6y=0\]
.<br /> We can write this as 
\[L(y)=0\]
  where 
\[L\]
  is the linear differential operator 
\[\frac{d^3 }{dx^3}-6 \frac{d^2}{dx^2}+11\frac{d}{dx}-6\]
.<br /> If we assume a solution of the form 
\[y=Ae^{\lambda x}\]
  the original equation becomes 
\[A \lambda^3 e^{\lambda x}- 6 A \lambda^2 e^{\lambda x}+11 A \lambda e^{\lambda x} -6A e^{\lambda x}=A e^{\lambda x}(\lambda^3-6 \lambda^2+11 \lambda -6)=0\]
.<br />
\[e^{\lambda x} \neq 0\]
  and 
\[A=0\]
  returns only the trivial solution 
\[y=0\]
, so 
\[\lambda^3-6 \lambda^2+11 \lambda -6=0\]
.<br /> This cubic factorises and we get 
\[(\lambda -1_)\lambda-2)(\lambda-3)=0\]
.<br /> Hence 
\[\lambda=1, \: 2, \: 3\]
  and the general solution is 
\[y=Ae^x+Be^{2x}+Ce^3x\]

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