Lecture Notes in Differential Equations - Bruce E. Shapiro

176 LESSON 20. THE WRONSKIAN
Multiply the first equation by y 2 and the second equation by y 1 ,
Subtracting the first from the second,
Substituting (20.45),
y 1 ′′ y 2 + p(t)y 1y ′ 2 + q(t)y 1 y 2 = 0 (20.48)
y 2 ′′ y 1 + p(t)y 2y ′ 1 + q(t)y 1 y 2 = 0 (20.49)
y 1 y ′′
2 − y 2 y ′′
1 + y 1 y ′ 2p(t) − y 2 y ′′
1 p(t) = 0 (20.50)
W ′ (t) = −p(t)W (t) (20.51)
This is a separable differential equation in W ; the solution is
( ∫ )
W (t) = Cexp − p(t)dt
(20.52)
This result is know is Abel’s Equation or Abel’s Formula, and we summarize
it in the following theorem.
Theorem 20.6. Abel’s Formula. Let y 1 and y 2 be solutions of
y ′′ + p(t)y ′ + q(t)y = 0 (20.53)
where p and q are continuous functions. Then for some constant C,
( ∫ )
W (y 1 , y 2 )(t) = Cexp − p(t)dt
(20.54)
Example 20.5. Find the Wronskian of
y ′′ − 2t sin(t 2 )y ′ + y sin t = 0 (20.55)
up to a constant multiple.
Using Abel’s equation,
(∫
W (t) = C exp
)
2t sin(t 2 )dt = Ce − cos t2 (20.56)
Note that as a consequence of Abel’s formula, the only way that W can be
zero is if C = 0; in this case, it is zero for all t. Thus the Wronskian of
two solutions of an ODE is either always zero or never zero. If
their Wronskian is never zero, by Theorem (20.4), the two solutions must
be linearly independent. On the other hand, if the Wronskian is zero at
some point t 0 then it is zero at all t, and
W (y 1 , y 2 )(t) =
∣ y 1(t) y 2 (t)
y 1(t)
′ y 2(t)
′ ∣ = 0 (20.57)