4: Second-Order ODEs with Constant Coefficients
The general linear second-order differential equation with independent variable \(t\) and dependent variable \(x = x(t)\) is given by \[\label{eq:1}\overset{..}{x}+p(t)\overset{.}{x}+q(t)x=g(t),\] where we have used the standard physics notation \(\overset{.}{x}= dx/dt\) and \(\overset{..}{x}= d^2x/dt^2\). A unique solution of \(\eqref{eq:1}\) requires initial values \(x(t_0) = x_0\) and \(\overset{.}{x}(t_0) = u_0\). 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 \(g(t) = 0\).