8: Second-Order ODEs, Constant Coefficients
Reference: Boyce and DiPrima, Chapter 3
The general linear second-order differential equation with independent variable \(t\) and dependent variable \(x=x(t)\) is given by
\[\ddot{x}+p(t) \dot{x}+q(t) x=g(t), \nonumber \]
where we have used the standard physics notation \(\dot{x}=d x / d t\) and \(\ddot{x}=d^{2} x / d t^{2}\) . Herein, we assume that \(p(t)\) and \(q(t)\) are continuous functions on the time interval for which we solve Equation \ref{8.1}. A unique solution of Equation \ref{8.1} requires initial values \(x\left(t_{0}\right)=x_{0}\) and \(\dot{x}\left(t_{0}\right)=u_{0}\) . 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 linear second-order ode is said to be homogeneous if \(g(t)=0\) .