SECOND-ORDER LINEAR DIFFERENTIAL EQUATION IN ITS EXACT FORM - Theoretical and Computational Research in the 21st Century - page 148

⎡ ⎤

⎛ ⎞

d d y

2

wp = Φ

⎢ ⎥

⎜ ⎟

(8)

dx dx wp ⎝ ⎠

⎣ ⎦

thus giving y ( x ) after integrating twice.

7.2 METHOD

Although Eq. (6) is not verified in general, it is possible to write Eq. (1) in
exact way. To do that, let us multiply this equation by

τ

ξ = (9)

wp

Therefore, Eq. (4) reduces to

pqr τ

''

++= τ

'

τ 0

(10)

which is precisely the homogeneous equation associated to Eq. (1); there-
fore, if

τ = yx

1 () is one of the two independent solutions:

py

''

++= qy

'

ry 0,

(11)

1 1 1

Eqs. (5) and (9) give:

'

y y y

α =

1

, β = -

1

, ξ =

1

(12)

w w w Φ

As a consequence, Eq. (3) takes its exact version:

⎡ ⎤

2

⎛⎞ =Φ

d y d y y

1 1

,

⎢ ⎥

⎜ ⎝⎠ (13)

dx w dx y wp

⎣ ⎦

1

and y ( x ) can be obtained after two integrations.
Equations (8) and (13) give the general solution of Eq. (1), in complete
harmony with the variation of parameters method developed by Lagrange
[2-4].