Question

Example of linear dependent differential equation

Answer 1

Example

The functions f(t) = 2sin2 t and g(t) = 1 - cos2(t) are linearly dependent since

(1)(2sin2 t) + (-2)(1 - cos2(t)) = 0

Example

The functions f(t) = t and g(t) = t2 are linearly independent since otherwise there would be nonzero constants c1 and c2 such that

c1t + c2t2 = 0

for all t. First let t = 1. Then

c1 + c2 = 0

Now let t = 2. Then

2c1 + 4c2 = 0

This is a system of 2 equations and two unknowns. The determinant of the corresponding matrix is

4 - 2 = 2

Since the determinant is nonzero, the only solution is the trivial solution. That is

c1 = c2 = 0

The two functions are linearly independent.