1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

16 Chapter 0 Ordinary Differential Equations

Inhomogeneity,f(t) Form of Trial Solution,up(t)

(a 0 tn+a 1 tn−^1 +···+an)eαt (A 0 tn+A 1 tn−^1 +···+An)eαt

(a 0 tn+···+an)eαtcos(βt)

+(b 0 tn+···+bn)eαtsin(βt)

(A 0 tn+···+An)eαtcos(βt)

+(B 0 tn+···+Bn)eαtsin(βt)

Table 4 Undetermined coefficients

generalsolutionofthegivenequationis

u(t)= 1 −^1

2

e−t+c 1 cos(t)+c 2 sin(t).

If two initial conditions are given, thenc 1 andc 2 are available to satisfy them.
Of course, an initial condition applies to the entire solution of the given dif-
ferential equation, not just touc(t).

Now we turn our attention to methods for finding particular solutions of
nonhomogeneous linear differential equations.

A. Undetermined Coefficients

This method involves guessing the form of a trial solution and then finding the
appropriate coefficients. Naturally, it is limited to the cases in which we can
guess successfully: when the equation has constant coefficients and the inho-
mogeneity is simple in form. Table 4 offers a summary of admissible inhomo-
geneities and the corresponding forms for particular solution. The parameters
n,α,βand the coefficientsa 0 ,...,an,b 0 ,...,bnare found by inspecting the
given inhomogeneity. The table compresses several cases. For instance,f(t)in
line 1 is a polynomial ifα=0oranexponentialifn=0andα=0. In line 2,
both sine and cosine must be included in the trial solution even if one is absent
fromf(t);butα=0 is allowed, and so isn=0.

Example.
Find a particular solution of

d^2 u

dt^2 +^5 u=te

−t.

Weuseline1ofTable4.Evidently,n=1andα=−1. The appropriate form
for the trial solution is

up(t)=(A 0 t+A 1 )e−t.

When we substitute this form into the differential equation, we obtain

(A 0 t+A 1 − 2 A 0 )e−t+ 5 (A 0 t+A 1 )e−t=te−t.