Homogeneous Linear ODE

Second-order linear ODEs with constant coefficients model every damped oscillator, every resonant circuit, and (in finance) many short-rate models. The homogeneous case is the foundation.

Solve the initial value problem

$y^{\prime \prime }-5y^{\prime }+6y=0,y(0)=1,y^{\prime }(0)=0.$

Give the explicit solution $y(t)$ and state which term dominates as $t\to \infty$.