## 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
$y=Ae^x+Be^{2x}+Ce^3x$