Second-order linear equation

What matters is that \(y\), \(y'\) and \(y''\) appear only to the first power, multiplied by functions of \(t\). When \(g(t)=0\) the equation is called homogeneous; if \(g\neq0\), nonhomogeneous, and \(g\) stands for an external force or source. The classification matters because almost every method in this topic depends on it.

Unlike first order, the general solution carries two arbitrary constants, because we must integrate twice. Two conditions are needed to fix them, for example the values of \(y\) and \(y'\) at a time \(t_0\). If \(p\), \(q\) and \(g\) are continuous on an interval, there is a unique solution that meets them.

When \(p\) and \(q\) are constants, we try \(y=e^{rt}\): substituting yields the polynomial \(r^2+pr+q=0\), the characteristic equation, and each root gives a solution. For example, \(y''-5y'+6y=0\) gives \(r=2\) and \(r=3\), so \(e^{2t}\) and \(e^{3t}\) are solutions.

A common mistake is to think any twice-differentiable function "looks like" a solution: you must substitute and check that the left side minus the right side vanishes. Another is calling \(y''+y^2=0\) linear; the square of the unknown makes it nonlinear. Next we see how solutions of the homogeneous version combine.