Solution
= Solution
Differentiate the <Wronskian> and cancel the mixed first-derivative terms:
$$
W'=u_1u_2''-u_2u_1''=u_1(-au_2'-bu_2)-u_2(-au_1'-bu_1)=-aW.
$$
This is the constant-coefficient form of <Abel's identity>. Hence $W(x)=Ce^{-ax}$, or $W(x)=W(x_0)e^{-a(x-x_0)}$. Since the <exponential function> is never zero, $W$ is either identically zero or nowhere zero on a connected interval of the homogeneous equation.