4: Second-Order ODEs with Constant Coefficients
( \newcommand{\kernel}{\mathrm{null}\,}\)
The general linear second-order differential equation with independent variable t and dependent variable x=x(t) is given by ..x+p(t).x+q(t)x=g(t),
where we have used the standard physics notation .x=dx/dt and ..x=d2x/dt2. A unique solution of (???) requires initial values x(t0)=x0 and .x(t0)=u0. The equation with constant coefficients—on which we will devote considerable effort— assumes that p(t) and q(t) are constants, independent of time. The second-order linear ode is said to be homogeneous if g(t)=0.