Skip to content
Resources

    Linearly Independent Solutions


    A set of functions f1(x), f2(x), ..., fn(x) is linearly independent on a ≤ x ≤ b, if constants c1, c2, ..., cn must equal zero in order to satsify:

    c1f1(x) + c2f2(x) + ... + cnfn(x) = 0

    on a ≤ x ≤ b.

    An nth order linear homogeneous differential equation always has n linearly independent solutions.  For linearly independent solutions represented by y1(x), y2(x), ..., yn(x), the general solution for the nth order linear equation is:

    y(x) = c1y1(x) + c2y2(x) + ... + cnyn(x)
    Example #1:
    Is the set of functions {1, x, sin x, 3sin x, cos x} linearly independent on [−1, 1]?

    Solution #1:
    The set of functions {1, x, sin x, 3sin x, cos x} is not linearly independent on [−1, 1] since 3sin x is a mulitple of sin x.

    Thus, selection of constants c1 = 0, c2 = 0, c3= 3, c4= −1, and c5 = 0 results in the following:

    (0) (1) + (0) (x) + (3) (sin x) + (−1)(3 sin x) + (0)(cos x) = 0

    which does not require selection of c1 = c2 = c3= c4= c5 = 0 in order to satisfy:
     
    c1f1(x) + c2f2(x) + c3f3(x)  + c4f4(x) + c5f5(x) = 0


    Example #2:
    Is the set of functions {1, x, sin x, ex, cos x} linearly independent on [−1, 1]?

    Solution #2:
    The set of functions {1, x, sin x, ex, cos x} is linearly independent on [−1, 1] as selection of c1 = c2 = c3= c4= c5 = 0 is required to satisfy:

    c1f1(x) + c2f2(x) + c3f3(x)  + c4f4(x) + c5f5(x) = 0

    Specifically,

    (0) (1) + (0) (x) + (0) (sin x) + (0) (ex) + (0)(cos x) = 0