Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 20
The Wronskian
We have seen that the sum of any two solutions y 1 , y 2 to
ay ′′ + by ′ + cy = 0 (20.1)
is also a solution, so a natural question becomes the following: how many
different solutions do we need to find to be certain that we have a general
solution? The answer is that every solution of (20.1) is a linear combination
of two linear independent solutions. In other words, if y 1
and y 2 are linearly independent (see definition (15.6)), i.e, there is no
possible combination of constants A and B, both nonzero, such that
Ay 1 (t) + By 2 (t) = 0 (20.2)
for all t, and if both y 1 and y 2 are solutions, then every solution of (20.1)
has the form
y = C 1 y 1 (t) + C 2 y 2 (t) (20.3)
We begin by considering the initial value problem 1
⎫
y ′′ + p(t)y ′ + q(t)y = 0 ⎪⎬
y(t 0 ) = y 0
⎪⎭
y ′ (t 0 ) = y 0
′
(20.4)
Suppose that y 1 (t) and y 2 (t) are both solutions of the homogeneous equation;
then
y(t) = Ay 1 (t) + By 2 (t) (20.5)
1 Eq. (20.1) can be put in the same form as (20.4) so long as a ≠ 0, by setting
p(t) = b/a and q(t) = c/a. However, (20.4) is considerably more general because we are
not requiring the coefficients to be constants.
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