4.3: The Wronskian
Suppose that having determined that two solutions of (4.2.1) are \(x = X_1(t)\) and \(x = X_2(t)\), we attempt to write the general solution to (4.2.1) as (4.2.2) . We must then ask whether this general solution will be able to satisfy the two initial conditions given by \[\label{eq:1}x(t_0)=x_0,\quad\overset{.}{x}(t_0)=u_0.\]
Applying these initial conditions to (4.2.2) , we obtain \[\begin{align}c_1X_1(t_0)+c_2X_2(t_0)&=x_0,\nonumber \\ c_1\overset{.}{X}_1(t_0)+c_2\overset{.}{X}_2(t_0)&=u_0,\label{eq:2}\end{align}\] ) which is observed to be a system of two linear equations for the two unknowns \(c_1\) and \(c_2\). Solution of \(\eqref{eq:2}\) by standard methods results in \[c_1=\frac{x_0\overset{.}{X}_2(t_0)-u_0X_2(t_0)}{W},\quad c_2=\frac{u_0X_1(t_0)-x_0\overset{.}{X}_1(t_0)}{W},\nonumber\] where \(W\) is called the Wronskian and is given by \[\label{eq:3}W=X_1(t_0)\overset{.}{X}_2(t_0)-\overset{.}{X}_1(t_0)X_2(t_0).\]
Evidently, the Wronskian must not be equal to zero \((W\neq 0)\) for a solution to exist.
For examples, the two solutions \[X_1(t)=A\sin\omega t,\quad X_2(t)=B\sin\omega t,\nonumber\] have a zero Wronskian at \(t=t_0\), as can be shown by computing \[\begin{aligned} W&=(A\sin\omega t_0)(B\omega\cos\omega t_0)-(A\omega\cos\omega t_0)(B\sin\omega t_0) \\ &=0;\end{aligned}\] while the two solutions \[X_1(t)=\sin\omega t,\quad X_2(t)=\cos\omega t,\nonumber\] with \(\omega\neq 0\), have a nonzero Wronskian at \(t=t_0\), \[\begin{aligned}W&=(\sin\omega t_0)(-\omega\sin\omega t_0)-(\omega\cos\omega t_0)(\cos\omega t_0) \\ &=-\omega.\end{aligned}\]
When the Wronskian is not equal to zero, we say that the two solutions \(X_1(t)\) and \(X_2(t)\) are linearly independent. The concept of linear independence is borrowed from linear algebra, and indeed, the set of all functions that satisfy (4.2.1) can be shown to form a two-dimensional vector space.