1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

20 Chapter 0 Ordinary Differential Equations

Notice that, astincreases, the terms that come from the complementary
solution approach 0, while the terms that come from the particular solution
persist. These cases are illustrated with animation on the CD.

B. Variation of Parameters

Generally, if a linear homogeneous differential equation can be solved, the cor-
responding nonhomogeneous equation can also be solved, at least in terms of
integrals.

1. First-order equations

Suppose thatuc(t)is a solution of the homogeneous equation

du

dt

=k(t)u. (5)

Then to find a particular solution of the nonhomogeneous equation

du

dt=k(t)u+f(t), (6)

we assume thatup(t)=v(t)uc(t). Substitutingupin this form into the differ-
ential equation (6) we have

dv

dtuc+v

duc

dt =k(t)vuc+f(t). (7)

However,u′c=k(t)uc,soonetermontheleftcancelsatermontheright,
leaving

dv

dt

uc=f(t), or dv

dt

= f(t)

uc(t)

. (8)

The latter is a nonhomogeneous equation of simplest type, which can be
solved forv(t)in one integration.

Example.
Use this method to find a solution of the nonhomogeneous equation

du

dt

= 5 u+t.

We should try the formup(t)=v(t)·e^5 t,becausee^5 tis a solution ofu′= 5 u.
Substituting the preceding form forup,wefind

dv

dt·e

5 t+v· 5 e 5 t= 5 ve 5 t+t,