on

a ≤

x ≤

b.

An

nth order linear homogeneous differential equation always has

n linearly independent solutions. For linearly independent solutions represented by

y_{1}(

x),

y_{2}(

x), ...,

y_{n}(

x), the general solution for the

nth order linear equation is:

y(x) = c_{1}y_{1}(x) + c_{2}y_{2}(x) + ... + c_{n}y_{n}(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

c_{1} = 0,

c_{2} = 0,

c_{3}= 3,

c_{4}=

−1, and c_{5} = 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

c_{1} =

c_{2} =

c_{3}=

c_{4}=

c_{5} = 0 in order to satisfy:

c_{1}f_{1}(x) + c_{2}f_{2}(x) + c_{3}f_{3}(x) + c_{4}f_{4}(x) + c_{5}f_{5}(x) = 0

The set of functions {1,

x, sin

x,

e^{x}, cos

x}

is linearly independent on [

−1, 1] as selection of c_{1} =

c_{2} =

c_{3}=

c_{4}=

c_{5} = 0 is required to satisfy:

c_{1}f_{1}(

x) +

c_{2}f_{2}(

x) +

c_{3}f_{3}(

x) +

c_{4}f_{4}(

x) +

c_{5}f_{5}(

x) = 0

Specifically,

(0) (1) + (0) (x) + (0) (sin x) + (0) (e^{x}) + (0)(cos x) = 0