Lecture Notes in Differential Equations - Bruce E. Shapiro

173
Definition 20.1. The Wronskian Determinant of two functions y 1 and
y 2 is given by
W (t) =
∣ y 1(t) y 2 (t)
y 1(t)
′ y 2(t)
′ ∣ = y 1(t)y 2(t) ′ − y 2 (t)y 1(t) ′ (20.18)
If y 1 and y 2 are a fundamental set of solutions of a differential equation,
then W (t) is called the Wronskian of the Differential Equation.
Example 20.1. Find the Wronskian of y 1 = sin t and y 2 = x 2 .
W (y 1 , y 2 )(t) = y 1 y ′ 2 − y 2 y ′ 1 (20.19)
= (sin t)(x 2 ) ′ − (x 2 )(sin t) ′ (20.20)
= 2x sin t − x 2 cos t (20.21)
Example 20.2. Find the Wronskian of the differential equation y ′′ −y = 0.
The roots of the characteristic equation is r 2 − 1 = 0 are ±1, and a fundamental
pair of solutions are y 1 = e t and y 2 = e −t . The Wronskian is
therefore
∣ W (x) =
∣ et e −t ∣∣∣
e t −e −t = −2. (20.22)
The discussion preceding Example (20.1) proved the following theorem.
Theorem 20.2. Existence of Solutions.
solutions of the equation
Let y 1 and y 2 be any two
such that
Then the initial value problem
has a solution, given by (20.17).
y ′′ + p(t)y ′ + q(t)y = 0 (20.23)
W (t 0 ) = y 1 (t 0 )y ′ 2(t 0 ) − y ′ 1(t 0 )y 2 (t 0 ) ≠ 0 (20.24)
y ′′ + p(t)y ′ + q(t)y = 0
⎫
⎪⎬
y(t 0 ) = y 0
⎪⎭
y ′ (t 0 ) = y 0
′
(20.25)
Theorem 20.3. General Solution. Suppose that y 1 and y 2 are solutions
of
y ′′ + p(t)y ′ + q(t)y = 0 (20.26)