An nth-order linear differential equation has the form bn(x)y(n)+bn−1(x)y(n−1)+…+b2(x)y′′+b1(x)y′+b0(x)y=g(x), where g(x) and all coefficients bj(x),j=0..n depend solely on the variable x. In other words, they do not depend on y or on any derivative of y.
A set of functions {y1(x),y2(x),…,yn(x)} is linearly dependent on a≤x≤b, if there exist constants c1, c2, ... , cn, not all zero, such that c1y1(x)+c2y2(x)+…+cnyn(x)≡0 on a≤x≤b.
Consider the nonhomogeneous differential equation y(n)+an−1(x)y(n−1)+…+a1(x)y′+a0(x)y=g(x).