1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

0.2 Nonhomogeneous Linear Equations 23

Now, Eq. (11) may be used to form a particular solution of the nonhomoge-
neous equation (9).
We m a y a l s o o b t a i nv 1 andv 2 by using definite integrals with variable upper
limit:

v 1 (t)=−

∫t

t 0

u 2 (z)f(z)

W(z) dz,v^2 (t)=

∫t

t 0

u 1 (z)f(z)

W(z) dz. (22)

Thelowerlimitisusuallytheinitialvalueoft,butmaybeanyconvenient
value. The particular solution can now be written as

up(t)=−u 1 (t)

∫t

t 0

u 2 (z)f(z)

W(z) dz+u^2 (t)

∫t

t 0

u 1 (z)f(z)

W(z) dz.

Furthermore, the factorsu 1 (t)andu 2 (t)can be inside the integrals (which are
notwith respect tot), and these can be combined to give a tidy formula, as
follows.

Theorem 3.Let u 1 (t)and u 2 (t)be independent solutions of

d^2 u

dt^2

+k(t)du

dt

+p(t)u=0(H)

with Wronskian W(t)=u 1 (t)u′ 2 (t)−u 2 (t)u′ 1 (t).Then

up(t)=

∫t

t 0

G(t,z)f(z)dz

is a particular solution of the nonhomogeneous equation

d^2 u

dt^2

+k(t)du

dt

+p(t)u=f(t), (NH)

where G is the Green’s function defined by

G(t,z)=u^1 (z)u^2 (tW)−(zu)^2 (z)u^1 (t). (23)

EXERCISES

In Exercises 1–10, find the general solution of the differential equation.