We can view equations (8.1) and (8.2) as a system of two linear equations with two unknowns c1 and c2. The determinant of such a system is the Wronskian

W(u, v) = uv′ − u′v . (8.3) If W(u, v) is nonzero for some value of x, then the only solution of the system of homogeneous linear equations is c1 = c2 = 0, and the functions are linearly independent. If, on the other hand, W(u, v) is zero for all values of x, then there are solutions for nonzero values of c1 and c2, and the functions are linearly dependent.