Lecture Notes in Differential Equations - Bruce E. Shapiro

248 LESSON 27. HIGHER ORDER EQUATIONS
where min(m ij ) is the minor of the ij th element. Differentiating,
dM
dt
n+1
∑
n+1
∑
= (−1) 1+i m ′ 1i min(m 1i ) + (−1) 1+i d
m 1i
dt min(m 1i) (27.170)
i=1
i=1
The first sum is d(M,1). Since (27.166) is true for any n × n matrix, we
can apply it to min(m 1i ) in the second sum.
dM
dt
= d(M, 1) + ∑ n+1
i=1 (−1)1+i m 1i
∑ n
j=1 d(min(m 1i), j)
(27.171)
which completes the inductive proof of the lemma.
Proof. (Abel’s Formula)
(n = 2). Suppose y 1 and y 2 are solutions of (27.165). Their Wronskian is
Differentiating,
W (t) = y 1 y ′ 2 − y 2 y ′ 1 (27.172)
W ′ (x) = y 1 y ′′
2 + y ′ 1y ′ 2 − y ′ 2y ′ 1 − y 2 y ′′
1 = y 1 y ′′
2 − y 2 y ′′
1 (27.173)
Since L 2 y 1 = L 2 y 2 = 0,
Hence
y ′′
1 = −p(t)y ′ 1 − q(t)y 1 (27.174)
y ′′
2 = −p(t)y ′ 2 − q(t)y 2 (27.175)
W ′ (t) = y 1 (−p(t)y 2 ′ − q(t)y 2 ) − y 2 (−p(t)y 1 ′ − q(t)y 1 )
= −p(t)(y 1 y 2 ′ − y 2 y 1)
′
= −p(t)W (t)
Rearranging and integrating gives W (t) = C exp [ − ∫ p(t)dt ] .
General Case. The Wronskian is
W [y 1 , ..., y n ](t) =
∣
y 1 · · · y n
.
.
y (n−1)
1 · · · y n
(n−1) ∣
(27.176)
(27.177)