ABSTRACT
A system of linear homogeneous differential equations is called adjoint to (A.1) if it has the form y′ = B(x)y, where B = −AT , and AT denotes the transpose of A.
A second-order differential equation can always be reduced to a system of first order equations. For example, the general second order differential equation
y′′ + a1(x) y′ + a2(x) y = 0, (A.3)
where ai, i = 1, 2, are real valued functions of x, is reduced to a system of first order equations by setting y = y1 and y′ = y2. Thus, (A.3) becomes
y′1 = y2,
y′2 = −a2y − a1y, or y′ = A(x)y, (A.4)
where A = [
0 1 −a2 −a1
]
. Since AT = [
0 −a2 1 −a1
]
, the adjoint system to (A.4) is y′ = −AT y, or
y′1 = a2(x) y2,
y′2 = −y1 + a1(x) y2.
