Chapter 51
Second order differential
equations of the form
a
d 2 y
dx 2 + b
dy
dx + cy = f (x)
51.1 Complementary function and
particular integral
If in the differential equation
a
d 2 y
dx 2 + b
d y
dx
+ cy = f (x)
(1)
the substitution y = u + v is made then:
a
d 2 (u + v)
dx 2
+ b
d(u + v)
dx
+ c(u + v) = f (x)
Rearranging gives:
a
d 2 u
dx 2 + b
du
dx
+ cu
+
a
d 2 v
dx 2 + b
dv
dx
+cv
= f (x)
If we let
a
d 2 v
dx 2 + b
dv
dx
+ cv = f (x)
(2)
then
a
d 2 u
dx 2 + b
du
dx
+ cu = 0
( 3 )
The general solution, u, of equation (3) will contain two
unknown constants, as required for the general solution
of equation (1). The method of solution of equation (3)
is shown in Chapter 50. The function u is called the
complementary function (C.F.).
If the particular solution, v, of equation (2) can be determined without containing any unknown constants then
y = u +v will give the general solution of equation (1).
The function v is called the particular integral (P.I.).
Hence the general solution of equation (1) is given by:
y = C.F. + P.I.
51.2 Procedure to solve differential
equations of the form
a
d 2 y
dx 2 + b
dy
dx
+ cy = f (x)
(i) Rewrite the given differential equation as
(aD 2 + bD+ c)y = f (x).
(ii) Substitute m for D, and solve the auxiliary
equation am 2 + bm +c = 0 for m.
(iii) Obtain the complementary function, u, which
is achieved using the same procedure as in
Section 50.2(c), page 478.