Lecture Notes in Differential Equations - Bruce E. Shapiro

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and therefore the system of equations
[ [ ]
y1 (t) y 2 A
y 1(t) ′ y 2(t)] ′ = 0 (20.58)
B
has a solution for A and B where at least one of A and B are non-zero.
This means that there exist A and B, at least one of which is non-zero,
such that
Ay 1 (t) + By 2 (t) = 0 (20.59)
Since this holds for all values of t, y 1 and y 2 are linearly dependent. This
proves the following theorem.
Theorem 20.7. Let y 1 and y 2 be solutions of
where p and q are continuous. Then
y ′′ + p(t)y ′ + q(t)y = 0 (20.60)
1. y 1 and y 2 are linearly dependent ⇐⇒ W (y 1 , y 2 )(t) = 0 for all t.
2. y 1 and y 2 are linearly independent ⇐⇒ W (y 1 , y 2 )(t) ≠ 0 for all t.
We can summarize our results about the Wronskian of solutions and Linear
Independence in the following theorem.
Theorem 20.8. Let y 1 (t) and y 2 (t) be solutions of
Then the following statements are equivalent:
y ′′ + p(t)y ′ + q(t)y = 0 (20.61)
1. y 1 and y 2 form a fundamental set of solutions.
2. y 1 and y 2 are linearly independent.
3. At some point t 0 , W (y 1 , y 2 )(t 0 ) ≠ 0.
4. W (y 1 , y 2 )(t) ≠ 0 for all t.