\[\frac{d^3 y}{dx^3}-6 \frac{d^2y}{dx^2}+11\frac{dy}{dx}-6y=0\]
.<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 \[y=Ae^x+Be^{2x}+Ce^3x\]